White Noise

set.seed(1010)
wn <- rnorm(300)
ts.plot(wn)

par(mfrow = c(1, 2)) 
acf(wn)
acf(wn, xlim = c(1, 20))

Proses MA(2)

Pembangkitan Proses MA(2)

Rumus MA (2): \[ Y_t = e_t - \theta_1 e_{t-1} - \theta_2 e_{t-2} \] dengan \(\theta_1\) = 0.4 dan \(\theta_2\) = 0.6.

set.seed(1010)
ma2 <- wn[1:2]
# Manual
for(i in 3:300){
  ma2[i] <- wn[i] + 0.4 * wn[i - 1] + 0.6 * wn[i - 2]
}
ma2
##   [1]  0.131540851 -0.897909868  1.071705175  0.422106613  0.690736960
##   [6]  1.501222304  2.311474041 -0.766756080 -0.375272150 -2.198788160
##  [11] -0.770197848 -1.235168383 -1.884954880 -1.148267311 -0.165788735
##  [16] -0.543028834 -0.698251506 -0.595470096 -1.361763778 -1.223980621
##  [21] -0.872896857  0.169333233 -0.629372146  0.504724047 -1.080874546
##  [26]  0.741720747 -0.434947966 -0.736240440 -0.898001000 -2.583414475
##  [31] -1.946601852  0.405487512 -0.001458042  1.425295218 -0.992023400
##  [36]  0.135076591 -0.756077274 -0.286595860 -0.890964944 -1.451956836
##  [41] -1.119449938 -0.332941093 -0.149323883  0.900367374 -0.471636090
##  [46]  0.291421594 -0.027813907 -0.360449808  0.298325813  1.103341959
##  [51]  2.564457298  0.752839686 -0.349149879 -1.291486161 -0.809197685
##  [56] -0.392505751  0.872787277  1.853672380  1.074961292  2.524086251
##  [61]  2.143785091  2.242967913  2.051877771  0.367790380  0.884919174
##  [66] -1.987698959  0.081619557 -1.399009877 -0.602541199 -0.573444638
##  [71]  0.085263990 -0.098161217 -0.549525949  0.557429691 -0.441211933
##  [76] -1.884837560 -1.408256548 -1.643425317 -0.199802210  2.034818368
##  [81]  2.074401582  0.006957924  1.779372840 -0.841885624  1.362601273
##  [86] -1.450510469 -0.907705729 -2.638564323 -0.122474746 -3.201485783
##  [91] -1.104120758 -1.141750432 -1.392526292 -0.166637697 -1.946759635
##  [96] -1.013813490 -1.976774025 -1.486695025  0.224089477 -1.855512321
## [101] -0.561068578 -0.999440381 -0.221545191  0.486811559 -0.036862815
## [106]  1.757290723  1.480190543  2.969062938 -0.326259776 -0.799197607
## [111] -0.808179007 -0.385273038 -0.348275098 -0.392513797 -0.063553296
## [116]  0.852222858 -0.496682531 -0.866590058 -0.905186937 -1.352029783
## [121] -0.713449120  0.331004139 -0.486937546  1.312935319 -1.237812436
## [126]  0.933383571  2.552545424  0.803360374  1.440802780 -0.446742946
## [131]  0.659236685  1.082173831  0.265199909  0.367789172  0.709282753
## [136]  1.385972415  0.802520637 -0.344503674  0.107120036  0.003476325
## [141]  1.392748099  0.779390228 -0.399078690 -1.094922224 -0.829390991
## [146] -1.864746160 -0.580276778 -0.478476828  0.356114144  1.850982956
## [151] -0.466901838 -1.164093416  0.137231410  1.168021742  3.261566866
## [156]  0.214870138  0.032633853 -1.860788790 -1.271352557 -0.825600567
## [161]  0.743515553  0.534708245 -0.005196627  0.202911024 -0.438505521
## [166]  1.111069126 -1.286172722  0.872794010  1.071890787  2.335583939
## [171]  0.940715791 -0.794767895 -1.429381201  0.145260459 -1.255209507
## [176]  0.938142085 -1.336776697 -0.115418604 -0.337151803 -0.660551833
## [181]  0.846697090 -0.042584924  1.938158720  2.904113441  0.118820814
## [186]  2.150886949  0.276600733  0.942818659 -0.463805594  1.786619578
## [191]  0.670057145  3.504060746  0.297509661  1.676409815  1.982706497
## [196]  1.634164958  1.189825747 -0.670367646 -0.652518236 -1.151377243
## [201]  0.148291269 -0.752651306  0.638619521  0.556678749  0.820734496
## [206]  0.804256928  0.552395492  1.165532369 -0.022253703  0.679273029
## [211] -0.342307569 -0.872145828 -1.478253847 -0.233706635  1.081041466
## [216] -0.788098167  0.819653163 -0.539672665  1.593565360  2.408443779
## [221]  0.221234597  2.239556708  1.617172807  1.181696461  0.784379850
## [226] -2.054508130 -0.569877189 -3.169308305 -0.651185696 -1.438204904
## [231]  0.072286160 -0.052327767 -0.096400102 -1.637654918 -2.208935588
## [236] -1.415831277 -0.444384665  0.850952959  1.501825451  1.038778453
## [241] -0.821503684 -0.975325820 -1.675504805 -1.246010938  1.860748958
## [246] -0.091680625  3.197791619  0.151798456  1.097527485  1.816289747
## [251]  0.956300101  0.770455741  0.294948362 -0.276326225  0.547203914
## [256]  0.016512894  1.018900124  1.300736390  2.087161732 -1.080885316
## [261] -1.642577893 -1.887157635  0.150056991  0.349669900 -0.797549589
## [266] -0.864764450 -1.211035093 -0.208354961  0.275473685 -1.066739767
## [271] -1.396369675 -1.512475371  0.406910154  0.214162508  0.575648631
## [276] -0.786718563 -0.674399863 -0.432780947  0.084417100 -0.397545750
## [281] -1.729390071 -1.769996549 -3.323988744 -2.402821052 -1.998021144
## [286] -0.912482741 -0.241228226 -0.802654706 -2.645049342 -1.464916821
## [291]  1.459673093 -0.056749942  0.358007119  1.119655842 -0.754682598
## [296]  1.260802819 -0.395196982 -0.532986680 -1.859035477 -3.405715799
# Dengan fungsi arima.sim
ma22 <- arima.sim(list(order = c(0,0,2), ma = c(0.4, 0.6)), n = 300)
ma22
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1]  1.071705175  0.422106613  0.690736960  1.501222304  2.311474041
##   [6] -0.766756080 -0.375272150 -2.198788160 -0.770197848 -1.235168383
##  [11] -1.884954880 -1.148267311 -0.165788735 -0.543028834 -0.698251506
##  [16] -0.595470096 -1.361763778 -1.223980621 -0.872896857  0.169333233
##  [21] -0.629372146  0.504724047 -1.080874546  0.741720747 -0.434947966
##  [26] -0.736240440 -0.898001000 -2.583414475 -1.946601852  0.405487512
##  [31] -0.001458042  1.425295218 -0.992023400  0.135076591 -0.756077274
##  [36] -0.286595860 -0.890964944 -1.451956836 -1.119449938 -0.332941093
##  [41] -0.149323883  0.900367374 -0.471636090  0.291421594 -0.027813907
##  [46] -0.360449808  0.298325813  1.103341959  2.564457298  0.752839686
##  [51] -0.349149879 -1.291486161 -0.809197685 -0.392505751  0.872787277
##  [56]  1.853672380  1.074961292  2.524086251  2.143785091  2.242967913
##  [61]  2.051877771  0.367790380  0.884919174 -1.987698959  0.081619557
##  [66] -1.399009877 -0.602541199 -0.573444638  0.085263990 -0.098161217
##  [71] -0.549525949  0.557429691 -0.441211933 -1.884837560 -1.408256548
##  [76] -1.643425317 -0.199802210  2.034818368  2.074401582  0.006957924
##  [81]  1.779372840 -0.841885624  1.362601273 -1.450510469 -0.907705729
##  [86] -2.638564323 -0.122474746 -3.201485783 -1.104120758 -1.141750432
##  [91] -1.392526292 -0.166637697 -1.946759635 -1.013813490 -1.976774025
##  [96] -1.486695025  0.224089477 -1.855512321 -0.561068578 -0.999440381
## [101] -0.221545191  0.486811559 -0.036862815  1.757290723  1.480190543
## [106]  2.969062938 -0.326259776 -0.799197607 -0.808179007 -0.385273038
## [111] -0.348275098 -0.392513797 -0.063553296  0.852222858 -0.496682531
## [116] -0.866590058 -0.905186937 -1.352029783 -0.713449120  0.331004139
## [121] -0.486937546  1.312935319 -1.237812436  0.933383571  2.552545424
## [126]  0.803360374  1.440802780 -0.446742946  0.659236685  1.082173831
## [131]  0.265199909  0.367789172  0.709282753  1.385972415  0.802520637
## [136] -0.344503674  0.107120036  0.003476325  1.392748099  0.779390228
## [141] -0.399078690 -1.094922224 -0.829390991 -1.864746160 -0.580276778
## [146] -0.478476828  0.356114144  1.850982956 -0.466901838 -1.164093416
## [151]  0.137231410  1.168021742  3.261566866  0.214870138  0.032633853
## [156] -1.860788790 -1.271352557 -0.825600567  0.743515553  0.534708245
## [161] -0.005196627  0.202911024 -0.438505521  1.111069126 -1.286172722
## [166]  0.872794010  1.071890787  2.335583939  0.940715791 -0.794767895
## [171] -1.429381201  0.145260459 -1.255209507  0.938142085 -1.336776697
## [176] -0.115418604 -0.337151803 -0.660551833  0.846697090 -0.042584924
## [181]  1.938158720  2.904113441  0.118820814  2.150886949  0.276600733
## [186]  0.942818659 -0.463805594  1.786619578  0.670057145  3.504060746
## [191]  0.297509661  1.676409815  1.982706497  1.634164958  1.189825747
## [196] -0.670367646 -0.652518236 -1.151377243  0.148291269 -0.752651306
## [201]  0.638619521  0.556678749  0.820734496  0.804256928  0.552395492
## [206]  1.165532369 -0.022253703  0.679273029 -0.342307569 -0.872145828
## [211] -1.478253847 -0.233706635  1.081041466 -0.788098167  0.819653163
## [216] -0.539672665  1.593565360  2.408443779  0.221234597  2.239556708
## [221]  1.617172807  1.181696461  0.784379850 -2.054508130 -0.569877189
## [226] -3.169308305 -0.651185696 -1.438204904  0.072286160 -0.052327767
## [231] -0.096400102 -1.637654918 -2.208935588 -1.415831277 -0.444384665
## [236]  0.850952959  1.501825451  1.038778453 -0.821503684 -0.975325820
## [241] -1.675504805 -1.246010938  1.860748958 -0.091680625  3.197791619
## [246]  0.151798456  1.097527485  1.816289747  0.956300101  0.770455741
## [251]  0.294948362 -0.276326225  0.547203914  0.016512894  1.018900124
## [256]  1.300736390  2.087161732 -1.080885316 -1.642577893 -1.887157635
## [261]  0.150056991  0.349669900 -0.797549589 -0.864764450 -1.211035093
## [266] -0.208354961  0.275473685 -1.066739767 -1.396369675 -1.512475371
## [271]  0.406910154  0.214162508  0.575648631 -0.786718563 -0.674399863
## [276] -0.432780947  0.084417100 -0.397545750 -1.729390071 -1.769996549
## [281] -3.323988744 -2.402821052 -1.998021144 -0.912482741 -0.241228226
## [286] -0.802654706 -2.645049342 -1.464916821  1.459673093 -0.056749942
## [291]  0.358007119  1.119655842 -0.754682598  1.260802819 -0.395196982
## [296] -0.532986680 -1.859035477 -3.405715799 -2.114086037 -0.915226670

Karakteristik MA(2)

Plot Time Series
ts.plot(ma2)

Berdasarkan plot time series, terlihat bahwa data MA(2) yang dibangkitkan stasioner dalam rataan.

Plot ACF
acf(ma2,lag.max = 20)

Berdasarkan plot ACF tersebut, terlihat bahwa plot ACF cuts off di lag pertama, tidak sejalan dengan teori yang ada.

Plot PACF
pacf(ma2)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cenderung tails off dan membentuk gelombang sinus.

Plot EACF
TSA::eacf(ma2)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x o o o o o o o o o  o  o  o 
## 1 x x x o o o o o o o o  o  o  o 
## 2 x x o x o o o o o o o  o  o  o 
## 3 x x o x o o o o o o o  o  o  o 
## 4 x x x x o o o o o o o  o  o  o 
## 5 x x x x o o o o o o o  o  o  o 
## 6 x x x x o o o o o o o  o  o  o 
## 7 x x x o o o o o o o o  o  o  o

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol pertama berada pada ordo AR(0) dan ordo MA(2).

Scatterplot Antar Lag

Yt dengan Yt-1
#Yt
yt_ma2 <- ma2[4:300]   
yt_ma2
##   [1]  0.422106613  0.690736960  1.501222304  2.311474041 -0.766756080
##   [6] -0.375272150 -2.198788160 -0.770197848 -1.235168383 -1.884954880
##  [11] -1.148267311 -0.165788735 -0.543028834 -0.698251506 -0.595470096
##  [16] -1.361763778 -1.223980621 -0.872896857  0.169333233 -0.629372146
##  [21]  0.504724047 -1.080874546  0.741720747 -0.434947966 -0.736240440
##  [26] -0.898001000 -2.583414475 -1.946601852  0.405487512 -0.001458042
##  [31]  1.425295218 -0.992023400  0.135076591 -0.756077274 -0.286595860
##  [36] -0.890964944 -1.451956836 -1.119449938 -0.332941093 -0.149323883
##  [41]  0.900367374 -0.471636090  0.291421594 -0.027813907 -0.360449808
##  [46]  0.298325813  1.103341959  2.564457298  0.752839686 -0.349149879
##  [51] -1.291486161 -0.809197685 -0.392505751  0.872787277  1.853672380
##  [56]  1.074961292  2.524086251  2.143785091  2.242967913  2.051877771
##  [61]  0.367790380  0.884919174 -1.987698959  0.081619557 -1.399009877
##  [66] -0.602541199 -0.573444638  0.085263990 -0.098161217 -0.549525949
##  [71]  0.557429691 -0.441211933 -1.884837560 -1.408256548 -1.643425317
##  [76] -0.199802210  2.034818368  2.074401582  0.006957924  1.779372840
##  [81] -0.841885624  1.362601273 -1.450510469 -0.907705729 -2.638564323
##  [86] -0.122474746 -3.201485783 -1.104120758 -1.141750432 -1.392526292
##  [91] -0.166637697 -1.946759635 -1.013813490 -1.976774025 -1.486695025
##  [96]  0.224089477 -1.855512321 -0.561068578 -0.999440381 -0.221545191
## [101]  0.486811559 -0.036862815  1.757290723  1.480190543  2.969062938
## [106] -0.326259776 -0.799197607 -0.808179007 -0.385273038 -0.348275098
## [111] -0.392513797 -0.063553296  0.852222858 -0.496682531 -0.866590058
## [116] -0.905186937 -1.352029783 -0.713449120  0.331004139 -0.486937546
## [121]  1.312935319 -1.237812436  0.933383571  2.552545424  0.803360374
## [126]  1.440802780 -0.446742946  0.659236685  1.082173831  0.265199909
## [131]  0.367789172  0.709282753  1.385972415  0.802520637 -0.344503674
## [136]  0.107120036  0.003476325  1.392748099  0.779390228 -0.399078690
## [141] -1.094922224 -0.829390991 -1.864746160 -0.580276778 -0.478476828
## [146]  0.356114144  1.850982956 -0.466901838 -1.164093416  0.137231410
## [151]  1.168021742  3.261566866  0.214870138  0.032633853 -1.860788790
## [156] -1.271352557 -0.825600567  0.743515553  0.534708245 -0.005196627
## [161]  0.202911024 -0.438505521  1.111069126 -1.286172722  0.872794010
## [166]  1.071890787  2.335583939  0.940715791 -0.794767895 -1.429381201
## [171]  0.145260459 -1.255209507  0.938142085 -1.336776697 -0.115418604
## [176] -0.337151803 -0.660551833  0.846697090 -0.042584924  1.938158720
## [181]  2.904113441  0.118820814  2.150886949  0.276600733  0.942818659
## [186] -0.463805594  1.786619578  0.670057145  3.504060746  0.297509661
## [191]  1.676409815  1.982706497  1.634164958  1.189825747 -0.670367646
## [196] -0.652518236 -1.151377243  0.148291269 -0.752651306  0.638619521
## [201]  0.556678749  0.820734496  0.804256928  0.552395492  1.165532369
## [206] -0.022253703  0.679273029 -0.342307569 -0.872145828 -1.478253847
## [211] -0.233706635  1.081041466 -0.788098167  0.819653163 -0.539672665
## [216]  1.593565360  2.408443779  0.221234597  2.239556708  1.617172807
## [221]  1.181696461  0.784379850 -2.054508130 -0.569877189 -3.169308305
## [226] -0.651185696 -1.438204904  0.072286160 -0.052327767 -0.096400102
## [231] -1.637654918 -2.208935588 -1.415831277 -0.444384665  0.850952959
## [236]  1.501825451  1.038778453 -0.821503684 -0.975325820 -1.675504805
## [241] -1.246010938  1.860748958 -0.091680625  3.197791619  0.151798456
## [246]  1.097527485  1.816289747  0.956300101  0.770455741  0.294948362
## [251] -0.276326225  0.547203914  0.016512894  1.018900124  1.300736390
## [256]  2.087161732 -1.080885316 -1.642577893 -1.887157635  0.150056991
## [261]  0.349669900 -0.797549589 -0.864764450 -1.211035093 -0.208354961
## [266]  0.275473685 -1.066739767 -1.396369675 -1.512475371  0.406910154
## [271]  0.214162508  0.575648631 -0.786718563 -0.674399863 -0.432780947
## [276]  0.084417100 -0.397545750 -1.729390071 -1.769996549 -3.323988744
## [281] -2.402821052 -1.998021144 -0.912482741 -0.241228226 -0.802654706
## [286] -2.645049342 -1.464916821  1.459673093 -0.056749942  0.358007119
## [291]  1.119655842 -0.754682598  1.260802819 -0.395196982 -0.532986680
## [296] -1.859035477 -3.405715799
#Yt-1
yt_1_ma2 <- ma2[3:299]
yt_1_ma2
##   [1]  1.071705175  0.422106613  0.690736960  1.501222304  2.311474041
##   [6] -0.766756080 -0.375272150 -2.198788160 -0.770197848 -1.235168383
##  [11] -1.884954880 -1.148267311 -0.165788735 -0.543028834 -0.698251506
##  [16] -0.595470096 -1.361763778 -1.223980621 -0.872896857  0.169333233
##  [21] -0.629372146  0.504724047 -1.080874546  0.741720747 -0.434947966
##  [26] -0.736240440 -0.898001000 -2.583414475 -1.946601852  0.405487512
##  [31] -0.001458042  1.425295218 -0.992023400  0.135076591 -0.756077274
##  [36] -0.286595860 -0.890964944 -1.451956836 -1.119449938 -0.332941093
##  [41] -0.149323883  0.900367374 -0.471636090  0.291421594 -0.027813907
##  [46] -0.360449808  0.298325813  1.103341959  2.564457298  0.752839686
##  [51] -0.349149879 -1.291486161 -0.809197685 -0.392505751  0.872787277
##  [56]  1.853672380  1.074961292  2.524086251  2.143785091  2.242967913
##  [61]  2.051877771  0.367790380  0.884919174 -1.987698959  0.081619557
##  [66] -1.399009877 -0.602541199 -0.573444638  0.085263990 -0.098161217
##  [71] -0.549525949  0.557429691 -0.441211933 -1.884837560 -1.408256548
##  [76] -1.643425317 -0.199802210  2.034818368  2.074401582  0.006957924
##  [81]  1.779372840 -0.841885624  1.362601273 -1.450510469 -0.907705729
##  [86] -2.638564323 -0.122474746 -3.201485783 -1.104120758 -1.141750432
##  [91] -1.392526292 -0.166637697 -1.946759635 -1.013813490 -1.976774025
##  [96] -1.486695025  0.224089477 -1.855512321 -0.561068578 -0.999440381
## [101] -0.221545191  0.486811559 -0.036862815  1.757290723  1.480190543
## [106]  2.969062938 -0.326259776 -0.799197607 -0.808179007 -0.385273038
## [111] -0.348275098 -0.392513797 -0.063553296  0.852222858 -0.496682531
## [116] -0.866590058 -0.905186937 -1.352029783 -0.713449120  0.331004139
## [121] -0.486937546  1.312935319 -1.237812436  0.933383571  2.552545424
## [126]  0.803360374  1.440802780 -0.446742946  0.659236685  1.082173831
## [131]  0.265199909  0.367789172  0.709282753  1.385972415  0.802520637
## [136] -0.344503674  0.107120036  0.003476325  1.392748099  0.779390228
## [141] -0.399078690 -1.094922224 -0.829390991 -1.864746160 -0.580276778
## [146] -0.478476828  0.356114144  1.850982956 -0.466901838 -1.164093416
## [151]  0.137231410  1.168021742  3.261566866  0.214870138  0.032633853
## [156] -1.860788790 -1.271352557 -0.825600567  0.743515553  0.534708245
## [161] -0.005196627  0.202911024 -0.438505521  1.111069126 -1.286172722
## [166]  0.872794010  1.071890787  2.335583939  0.940715791 -0.794767895
## [171] -1.429381201  0.145260459 -1.255209507  0.938142085 -1.336776697
## [176] -0.115418604 -0.337151803 -0.660551833  0.846697090 -0.042584924
## [181]  1.938158720  2.904113441  0.118820814  2.150886949  0.276600733
## [186]  0.942818659 -0.463805594  1.786619578  0.670057145  3.504060746
## [191]  0.297509661  1.676409815  1.982706497  1.634164958  1.189825747
## [196] -0.670367646 -0.652518236 -1.151377243  0.148291269 -0.752651306
## [201]  0.638619521  0.556678749  0.820734496  0.804256928  0.552395492
## [206]  1.165532369 -0.022253703  0.679273029 -0.342307569 -0.872145828
## [211] -1.478253847 -0.233706635  1.081041466 -0.788098167  0.819653163
## [216] -0.539672665  1.593565360  2.408443779  0.221234597  2.239556708
## [221]  1.617172807  1.181696461  0.784379850 -2.054508130 -0.569877189
## [226] -3.169308305 -0.651185696 -1.438204904  0.072286160 -0.052327767
## [231] -0.096400102 -1.637654918 -2.208935588 -1.415831277 -0.444384665
## [236]  0.850952959  1.501825451  1.038778453 -0.821503684 -0.975325820
## [241] -1.675504805 -1.246010938  1.860748958 -0.091680625  3.197791619
## [246]  0.151798456  1.097527485  1.816289747  0.956300101  0.770455741
## [251]  0.294948362 -0.276326225  0.547203914  0.016512894  1.018900124
## [256]  1.300736390  2.087161732 -1.080885316 -1.642577893 -1.887157635
## [261]  0.150056991  0.349669900 -0.797549589 -0.864764450 -1.211035093
## [266] -0.208354961  0.275473685 -1.066739767 -1.396369675 -1.512475371
## [271]  0.406910154  0.214162508  0.575648631 -0.786718563 -0.674399863
## [276] -0.432780947  0.084417100 -0.397545750 -1.729390071 -1.769996549
## [281] -3.323988744 -2.402821052 -1.998021144 -0.912482741 -0.241228226
## [286] -0.802654706 -2.645049342 -1.464916821  1.459673093 -0.056749942
## [291]  0.358007119  1.119655842 -0.754682598  1.260802819 -0.395196982
## [296] -0.532986680 -1.859035477
plot(y=yt_ma2,x=yt_1_ma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).

Yt dengan Yt-2
#Yt-2
yt_2_ma2 <- ma2[2:298]
yt_2_ma2
##   [1] -0.897909868  1.071705175  0.422106613  0.690736960  1.501222304
##   [6]  2.311474041 -0.766756080 -0.375272150 -2.198788160 -0.770197848
##  [11] -1.235168383 -1.884954880 -1.148267311 -0.165788735 -0.543028834
##  [16] -0.698251506 -0.595470096 -1.361763778 -1.223980621 -0.872896857
##  [21]  0.169333233 -0.629372146  0.504724047 -1.080874546  0.741720747
##  [26] -0.434947966 -0.736240440 -0.898001000 -2.583414475 -1.946601852
##  [31]  0.405487512 -0.001458042  1.425295218 -0.992023400  0.135076591
##  [36] -0.756077274 -0.286595860 -0.890964944 -1.451956836 -1.119449938
##  [41] -0.332941093 -0.149323883  0.900367374 -0.471636090  0.291421594
##  [46] -0.027813907 -0.360449808  0.298325813  1.103341959  2.564457298
##  [51]  0.752839686 -0.349149879 -1.291486161 -0.809197685 -0.392505751
##  [56]  0.872787277  1.853672380  1.074961292  2.524086251  2.143785091
##  [61]  2.242967913  2.051877771  0.367790380  0.884919174 -1.987698959
##  [66]  0.081619557 -1.399009877 -0.602541199 -0.573444638  0.085263990
##  [71] -0.098161217 -0.549525949  0.557429691 -0.441211933 -1.884837560
##  [76] -1.408256548 -1.643425317 -0.199802210  2.034818368  2.074401582
##  [81]  0.006957924  1.779372840 -0.841885624  1.362601273 -1.450510469
##  [86] -0.907705729 -2.638564323 -0.122474746 -3.201485783 -1.104120758
##  [91] -1.141750432 -1.392526292 -0.166637697 -1.946759635 -1.013813490
##  [96] -1.976774025 -1.486695025  0.224089477 -1.855512321 -0.561068578
## [101] -0.999440381 -0.221545191  0.486811559 -0.036862815  1.757290723
## [106]  1.480190543  2.969062938 -0.326259776 -0.799197607 -0.808179007
## [111] -0.385273038 -0.348275098 -0.392513797 -0.063553296  0.852222858
## [116] -0.496682531 -0.866590058 -0.905186937 -1.352029783 -0.713449120
## [121]  0.331004139 -0.486937546  1.312935319 -1.237812436  0.933383571
## [126]  2.552545424  0.803360374  1.440802780 -0.446742946  0.659236685
## [131]  1.082173831  0.265199909  0.367789172  0.709282753  1.385972415
## [136]  0.802520637 -0.344503674  0.107120036  0.003476325  1.392748099
## [141]  0.779390228 -0.399078690 -1.094922224 -0.829390991 -1.864746160
## [146] -0.580276778 -0.478476828  0.356114144  1.850982956 -0.466901838
## [151] -1.164093416  0.137231410  1.168021742  3.261566866  0.214870138
## [156]  0.032633853 -1.860788790 -1.271352557 -0.825600567  0.743515553
## [161]  0.534708245 -0.005196627  0.202911024 -0.438505521  1.111069126
## [166] -1.286172722  0.872794010  1.071890787  2.335583939  0.940715791
## [171] -0.794767895 -1.429381201  0.145260459 -1.255209507  0.938142085
## [176] -1.336776697 -0.115418604 -0.337151803 -0.660551833  0.846697090
## [181] -0.042584924  1.938158720  2.904113441  0.118820814  2.150886949
## [186]  0.276600733  0.942818659 -0.463805594  1.786619578  0.670057145
## [191]  3.504060746  0.297509661  1.676409815  1.982706497  1.634164958
## [196]  1.189825747 -0.670367646 -0.652518236 -1.151377243  0.148291269
## [201] -0.752651306  0.638619521  0.556678749  0.820734496  0.804256928
## [206]  0.552395492  1.165532369 -0.022253703  0.679273029 -0.342307569
## [211] -0.872145828 -1.478253847 -0.233706635  1.081041466 -0.788098167
## [216]  0.819653163 -0.539672665  1.593565360  2.408443779  0.221234597
## [221]  2.239556708  1.617172807  1.181696461  0.784379850 -2.054508130
## [226] -0.569877189 -3.169308305 -0.651185696 -1.438204904  0.072286160
## [231] -0.052327767 -0.096400102 -1.637654918 -2.208935588 -1.415831277
## [236] -0.444384665  0.850952959  1.501825451  1.038778453 -0.821503684
## [241] -0.975325820 -1.675504805 -1.246010938  1.860748958 -0.091680625
## [246]  3.197791619  0.151798456  1.097527485  1.816289747  0.956300101
## [251]  0.770455741  0.294948362 -0.276326225  0.547203914  0.016512894
## [256]  1.018900124  1.300736390  2.087161732 -1.080885316 -1.642577893
## [261] -1.887157635  0.150056991  0.349669900 -0.797549589 -0.864764450
## [266] -1.211035093 -0.208354961  0.275473685 -1.066739767 -1.396369675
## [271] -1.512475371  0.406910154  0.214162508  0.575648631 -0.786718563
## [276] -0.674399863 -0.432780947  0.084417100 -0.397545750 -1.729390071
## [281] -1.769996549 -3.323988744 -2.402821052 -1.998021144 -0.912482741
## [286] -0.241228226 -0.802654706 -2.645049342 -1.464916821  1.459673093
## [291] -0.056749942  0.358007119  1.119655842 -0.754682598  1.260802819
## [296] -0.395196982 -0.532986680
plot(y=yt_ma2,x=yt_2_ma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).

Yt dengan Yt-3
#Yt-3
yt_3_ma2 <- ma2[1:297]
yt_3_ma2
##   [1]  0.131540851 -0.897909868  1.071705175  0.422106613  0.690736960
##   [6]  1.501222304  2.311474041 -0.766756080 -0.375272150 -2.198788160
##  [11] -0.770197848 -1.235168383 -1.884954880 -1.148267311 -0.165788735
##  [16] -0.543028834 -0.698251506 -0.595470096 -1.361763778 -1.223980621
##  [21] -0.872896857  0.169333233 -0.629372146  0.504724047 -1.080874546
##  [26]  0.741720747 -0.434947966 -0.736240440 -0.898001000 -2.583414475
##  [31] -1.946601852  0.405487512 -0.001458042  1.425295218 -0.992023400
##  [36]  0.135076591 -0.756077274 -0.286595860 -0.890964944 -1.451956836
##  [41] -1.119449938 -0.332941093 -0.149323883  0.900367374 -0.471636090
##  [46]  0.291421594 -0.027813907 -0.360449808  0.298325813  1.103341959
##  [51]  2.564457298  0.752839686 -0.349149879 -1.291486161 -0.809197685
##  [56] -0.392505751  0.872787277  1.853672380  1.074961292  2.524086251
##  [61]  2.143785091  2.242967913  2.051877771  0.367790380  0.884919174
##  [66] -1.987698959  0.081619557 -1.399009877 -0.602541199 -0.573444638
##  [71]  0.085263990 -0.098161217 -0.549525949  0.557429691 -0.441211933
##  [76] -1.884837560 -1.408256548 -1.643425317 -0.199802210  2.034818368
##  [81]  2.074401582  0.006957924  1.779372840 -0.841885624  1.362601273
##  [86] -1.450510469 -0.907705729 -2.638564323 -0.122474746 -3.201485783
##  [91] -1.104120758 -1.141750432 -1.392526292 -0.166637697 -1.946759635
##  [96] -1.013813490 -1.976774025 -1.486695025  0.224089477 -1.855512321
## [101] -0.561068578 -0.999440381 -0.221545191  0.486811559 -0.036862815
## [106]  1.757290723  1.480190543  2.969062938 -0.326259776 -0.799197607
## [111] -0.808179007 -0.385273038 -0.348275098 -0.392513797 -0.063553296
## [116]  0.852222858 -0.496682531 -0.866590058 -0.905186937 -1.352029783
## [121] -0.713449120  0.331004139 -0.486937546  1.312935319 -1.237812436
## [126]  0.933383571  2.552545424  0.803360374  1.440802780 -0.446742946
## [131]  0.659236685  1.082173831  0.265199909  0.367789172  0.709282753
## [136]  1.385972415  0.802520637 -0.344503674  0.107120036  0.003476325
## [141]  1.392748099  0.779390228 -0.399078690 -1.094922224 -0.829390991
## [146] -1.864746160 -0.580276778 -0.478476828  0.356114144  1.850982956
## [151] -0.466901838 -1.164093416  0.137231410  1.168021742  3.261566866
## [156]  0.214870138  0.032633853 -1.860788790 -1.271352557 -0.825600567
## [161]  0.743515553  0.534708245 -0.005196627  0.202911024 -0.438505521
## [166]  1.111069126 -1.286172722  0.872794010  1.071890787  2.335583939
## [171]  0.940715791 -0.794767895 -1.429381201  0.145260459 -1.255209507
## [176]  0.938142085 -1.336776697 -0.115418604 -0.337151803 -0.660551833
## [181]  0.846697090 -0.042584924  1.938158720  2.904113441  0.118820814
## [186]  2.150886949  0.276600733  0.942818659 -0.463805594  1.786619578
## [191]  0.670057145  3.504060746  0.297509661  1.676409815  1.982706497
## [196]  1.634164958  1.189825747 -0.670367646 -0.652518236 -1.151377243
## [201]  0.148291269 -0.752651306  0.638619521  0.556678749  0.820734496
## [206]  0.804256928  0.552395492  1.165532369 -0.022253703  0.679273029
## [211] -0.342307569 -0.872145828 -1.478253847 -0.233706635  1.081041466
## [216] -0.788098167  0.819653163 -0.539672665  1.593565360  2.408443779
## [221]  0.221234597  2.239556708  1.617172807  1.181696461  0.784379850
## [226] -2.054508130 -0.569877189 -3.169308305 -0.651185696 -1.438204904
## [231]  0.072286160 -0.052327767 -0.096400102 -1.637654918 -2.208935588
## [236] -1.415831277 -0.444384665  0.850952959  1.501825451  1.038778453
## [241] -0.821503684 -0.975325820 -1.675504805 -1.246010938  1.860748958
## [246] -0.091680625  3.197791619  0.151798456  1.097527485  1.816289747
## [251]  0.956300101  0.770455741  0.294948362 -0.276326225  0.547203914
## [256]  0.016512894  1.018900124  1.300736390  2.087161732 -1.080885316
## [261] -1.642577893 -1.887157635  0.150056991  0.349669900 -0.797549589
## [266] -0.864764450 -1.211035093 -0.208354961  0.275473685 -1.066739767
## [271] -1.396369675 -1.512475371  0.406910154  0.214162508  0.575648631
## [276] -0.786718563 -0.674399863 -0.432780947  0.084417100 -0.397545750
## [281] -1.729390071 -1.769996549 -3.323988744 -2.402821052 -1.998021144
## [286] -0.912482741 -0.241228226 -0.802654706 -2.645049342 -1.464916821
## [291]  1.459673093 -0.056749942  0.358007119  1.119655842 -0.754682598
## [296]  1.260802819 -0.395196982
plot(y=yt_ma2,x=yt_3_ma2)

Berdasarkan scatterplot tersebut, terlihat bahwa cenderung tidak terdapat hubungan antara \(Y_t\) dan \(Y_{t-3}\).

Korelasi Antar Lag

Data Bangkitan
cor(yt_ma2,yt_1_ma2)
## [1] 0.4301445

Nalai korelasi antara \(Y_t\) dan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:

\[ \rho_{1} = \frac{-\theta_1 + \theta_1\theta_2}{1 + (-\theta_1)^2 + (-\theta_2)^2} = \frac{-(-0.4) + (-0.4)(-0.6)}{1 + (-0.4)^2 + (-0.6)^2} \approx 0.421 \]

cor(yt_ma2,yt_2_ma2)
## [1] 0.3820866

Nilai korelasi antara \(Y_t\) dan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:

\[ \rho_{2} = \frac{-\theta_2}{1 + (-\theta_1)^2 + (-\theta_2)^2} = \frac{-(-0.6)}{1 + (-0.4)^2 + (-0.6)^2} \approx 0.395 \]

cor(yt_ma2,yt_3_ma2)
## [1] 0.05579518

Nilai korelasi antara \(Y_t\) dan \(Y_{t-3}\) dari hasil simulasi kurang mendekati teori yang ada yaitu 0.

Kesimpulan: Meskipun nilai autokorelasi empiris cukup dekat dengan nilai teoritis, masih terdapat perbedaan pola, misalnya pada \(\rho_3\) yang secara teoritis seharusnya bernilai nol. Ketidaksesuaian ini umumnya disebabkan oleh keterbatasan sampel, fluktuasi estimasi ACF, atau parameter yang relatif kecil sehingga kontribusi lag tertentu tampak lemah. Dengan demikian, model MA(2) tetap konsisten dengan struktur teoritis, sedangkan perbedaan kecil yang muncul lebih disebabkan oleh faktor acak dalam proses estimasi, bukan karena kesalahan dalam pemilihan model.

Proses AR(2)

Pembangkitan Proses AR(2)

Rumus AR (2): \[ Y_t = \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + e_t \] dengan \(\phi_1\) = 0.5 dan \(\phi_2\) = 0.2.

set.seed(1010)
n<-length(wn)
# Manual
ar2 <- wn[1:2]
for (i in 3:n) {
  ar2[i] <- wn[i] + 0.5 * ar2[i-1] + 0.2 * ar2[i-2]
}
ar2
##   [1]  0.131540851 -0.897909868  0.929297848  0.705141640  0.249970706
##   [6]  1.630575045  2.804006969 -0.632864527 -0.349744995 -0.845998100
##  [11] -0.688856343 -1.343701238 -2.245002413 -1.467312487 -0.456788733
##  [16] -1.309590275 -1.564831822 -0.839692440 -1.685225610 -1.976351619
##  [21] -1.240349843 -0.300580252 -1.364600891 -0.280115127 -1.099028358
##  [26]  0.133222967 -0.472015853 -1.261331210 -1.010989626 -2.595626463
##  [31] -2.539914335  0.135085568 -0.587623429  0.062873138 -1.121668605
##  [36] -0.196751533 -0.598041725 -0.725739020 -1.053296316 -1.663004703
##  [41] -1.422636980 -0.629945133 -0.686128345  0.217580950 -0.722747914
##  [46] -0.160691574  0.101011878 -0.566744798  0.073656410  1.243158084
##  [51]  2.470800226  0.711268152 -0.290944268 -0.374751162 -0.221706023
##  [56] -0.364932764  0.703316452  1.867772149  0.955796044  2.469579745
##  [61]  2.993735248  2.635760644  2.769842614  1.551591476  1.846947073
##  [66] -0.744491076  0.559766909 -0.306104106 -0.806377951 -0.469489115
##  [71]  0.150443138 -0.332375194 -0.888023314  0.535932096 -0.318263458
##  [76] -2.401184208 -1.487630879 -1.368579826 -0.989775631  1.356114931
##  [81]  1.709392162 -0.633670165  1.770677990  0.274219021  1.000219974
##  [86] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
##  [91] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
##  [96] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
## [101] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
## [106]  1.264051868  1.430515867  2.598609523  0.061890650 -0.617448193
## [111]  0.276831567  0.101251347 -0.620719712 -0.443733826  0.087912564
## [116]  0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
## [121] -1.231732070  0.123787254 -0.817652540  0.574641597 -1.117586460
## [126]  0.410843111  2.937392529  0.659205141  0.941254795  0.681051840
## [131]  1.142075967  1.496952499  0.558230240  0.639936785  1.367521449
## [136]  1.786498994  1.017829854 -0.003569923  0.746169264  0.679962460
## [141]  1.432293504  1.069743017 -0.230637043 -0.706067505 -0.275492766
## [146] -1.710354850 -0.992195806 -0.425040791 -0.170944748  1.336619651
## [151] -0.579629920 -1.605344237  0.580025068  1.487228085  2.614698261
## [156]  0.206662565  0.165110137 -0.713557580 -0.983434777 -0.693691169
## [161]  0.619494728  0.276063826 -0.483104007  0.251552914 -0.137485807
## [166]  0.896558099 -1.131402124  0.558283920  1.678193384  2.069400658
## [171]  0.888396815 -0.415100789 -0.660815187  0.748117966 -1.099365355
## [176]  0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [181]  0.468932148  0.020681103  1.499110867  3.126409313  0.195770249
## [186]  2.117325973  1.817108123  1.150626089  0.115911690  2.512665732
## [191]  1.553432269  3.338992399  1.289444828  2.029379968  3.382984512
## [196]  2.457255539  1.684852166  0.535726173  0.403799194 -0.283028070
## [201]  0.444986829 -0.433816175  0.447145774  0.823288188  0.702182433
## [206]  0.827674987  0.861233509  1.451744630  0.349452445  0.850459079
## [211]  0.327858831 -0.702450476 -1.248961917  0.008528714  1.104098600
## [216] -1.238293635  0.328369561  0.161403931  1.205972092  2.472945399
## [221]  0.328072560  2.255412751  2.861535473  1.438188479  1.252317156
## [226] -0.858881100 -0.016332851 -2.350702233 -1.059086108 -1.183245283
## [231] -0.729447937 -0.573160493 -0.584550104 -2.000657480 -2.597426055
## [236] -1.566350686 -0.911930377 -0.154070741  0.761900065  0.611286982
## [241] -1.080733526 -0.934623033 -1.229103342 -1.519320967  1.469792754
## [246] -0.220051190  2.156993581  0.787710783  0.837655525  2.535764561
## [251]  1.600514779  0.836188700  1.122573472  0.581180746  0.890620773
## [256]  0.516259386  1.247959408  1.730458037  2.313668714 -0.659434350
## [261] -1.363925887 -0.804814278  0.369415933 -0.049849610 -1.345918269
## [266] -0.945586014 -1.011022981 -0.637770952 -0.106931692 -1.447537800
## [271] -1.883412836 -1.528475651  0.067848989 -0.362758876 -0.281027342
## [276] -0.899903984 -0.837891287 -0.486910734 -0.172382511 -0.743789022
## [281] -2.054865504 -1.950659938 -3.411407367 -3.223933375 -2.625967645
## [286] -2.060706822 -1.556568946 -1.930909946 -3.625011258 -2.280016976
## [291]  1.036139594 -1.106346059 -1.261263207  1.334929741 -0.665011734
## [296]  0.315274152  0.125254497 -0.676028950 -1.911686502 -3.376249052
# Dengan fungsi arima.sim
ar22 <- arima.sim(list(order = c(2,0,0), ar = c(0.5, 0.2)),n = 200)
ar22
## Time Series:
## Start = 1 
## End = 200 
## Frequency = 1 
##   [1]  0.133295622 -0.471960465 -1.261288985 -1.010957436 -2.595601924
##   [6] -2.539895627  0.135099830 -0.587612557  0.062881427 -1.121662286
##  [11] -0.196746715 -0.598038053 -0.725736220 -1.053294182 -1.663003076
##  [16] -1.422635739 -0.629944187 -0.686127624  0.217581499 -0.722747495
##  [21] -0.160691255  0.101012122 -0.566744612  0.073656551  1.243158192
##  [26]  2.470800309  0.711268215 -0.290944221 -0.374751126 -0.221705995
##  [31] -0.364932743  0.703316468  1.867772162  0.955796053  2.469579753
##  [36]  2.993735253  2.635760648  2.769842617  1.551591479  1.846947075
##  [41] -0.744491075  0.559766910 -0.306104105 -0.806377951 -0.469489115
##  [46]  0.150443138 -0.332375194 -0.888023314  0.535932097 -0.318263458
##  [51] -2.401184208 -1.487630879 -1.368579826 -0.989775631  1.356114931
##  [56]  1.709392163 -0.633670165  1.770677990  0.274219021  1.000219974
##  [61] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
##  [66] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
##  [71] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
##  [76] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
##  [81]  1.264051868  1.430515867  2.598609523  0.061890650 -0.617448193
##  [86]  0.276831567  0.101251347 -0.620719712 -0.443733826  0.087912564
##  [91]  0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
##  [96] -1.231732070  0.123787254 -0.817652540  0.574641597 -1.117586460
## [101]  0.410843111  2.937392529  0.659205141  0.941254795  0.681051840
## [106]  1.142075967  1.496952499  0.558230240  0.639936785  1.367521449
## [111]  1.786498994  1.017829854 -0.003569923  0.746169264  0.679962460
## [116]  1.432293504  1.069743017 -0.230637043 -0.706067505 -0.275492766
## [121] -1.710354850 -0.992195806 -0.425040791 -0.170944748  1.336619651
## [126] -0.579629920 -1.605344237  0.580025068  1.487228085  2.614698261
## [131]  0.206662565  0.165110137 -0.713557580 -0.983434777 -0.693691169
## [136]  0.619494728  0.276063826 -0.483104007  0.251552914 -0.137485807
## [141]  0.896558099 -1.131402124  0.558283920  1.678193384  2.069400658
## [146]  0.888396815 -0.415100789 -0.660815187  0.748117966 -1.099365355
## [151]  0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [156]  0.468932148  0.020681103  1.499110867  3.126409313  0.195770249
## [161]  2.117325973  1.817108123  1.150626089  0.115911690  2.512665732
## [166]  1.553432269  3.338992399  1.289444828  2.029379968  3.382984512
## [171]  2.457255539  1.684852166  0.535726173  0.403799194 -0.283028070
## [176]  0.444986829 -0.433816175  0.447145774  0.823288188  0.702182433
## [181]  0.827674987  0.861233509  1.451744630  0.349452445  0.850459079
## [186]  0.327858831 -0.702450476 -1.248961917  0.008528714  1.104098600
## [191] -1.238293635  0.328369561  0.161403931  1.205972092  2.472945399
## [196]  0.328072560  2.255412751  2.861535473  1.438188479  1.252317156

Karakteristik AR(2)

Plot Time Series
ts.plot(ar2)

Berdasarkan plot time series tersebut terlihat bahwa data cenderung stasioner pada rataan.

Plot ACF
acf(ar2)

Berdasarkan plot ACF tersebut terlihat bahwa plot ACF cenderung tails off dan cenderung membentuk pola grafik sinus.

PLot PACF
pacf(ar2)

Berdasarkan plot PACF tersebut, terlihat bahwa plot PACF cuts off pada lag kedua, sejalan dengan teori yang ada.

Plot EACF
TSA::eacf(ar2)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x o o  o  o  o 
## 1 x o o o o o o o o o o  o  o  o 
## 2 x o o o o o o o o o o  o  o  o 
## 3 o x o o o o o o o o o  o  o  o 
## 4 o x o o o o o o o o o  o  o  o 
## 5 x x o o o o o o o o o  o  o  o 
## 6 x x x o x o o o o o o  o  o  o 
## 7 x x x x x o o o o o o  o  o  o

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(1).

Scatterplot Antar Lag

Yt dengan Yt-1
#Yt
yt_ar2 <- ar2[4:300]   
yt_ar2
##   [1]  0.705141640  0.249970706  1.630575045  2.804006969 -0.632864527
##   [6] -0.349744995 -0.845998100 -0.688856343 -1.343701238 -2.245002413
##  [11] -1.467312487 -0.456788733 -1.309590275 -1.564831822 -0.839692440
##  [16] -1.685225610 -1.976351619 -1.240349843 -0.300580252 -1.364600891
##  [21] -0.280115127 -1.099028358  0.133222967 -0.472015853 -1.261331210
##  [26] -1.010989626 -2.595626463 -2.539914335  0.135085568 -0.587623429
##  [31]  0.062873138 -1.121668605 -0.196751533 -0.598041725 -0.725739020
##  [36] -1.053296316 -1.663004703 -1.422636980 -0.629945133 -0.686128345
##  [41]  0.217580950 -0.722747914 -0.160691574  0.101011878 -0.566744798
##  [46]  0.073656410  1.243158084  2.470800226  0.711268152 -0.290944268
##  [51] -0.374751162 -0.221706023 -0.364932764  0.703316452  1.867772149
##  [56]  0.955796044  2.469579745  2.993735248  2.635760644  2.769842614
##  [61]  1.551591476  1.846947073 -0.744491076  0.559766909 -0.306104106
##  [66] -0.806377951 -0.469489115  0.150443138 -0.332375194 -0.888023314
##  [71]  0.535932096 -0.318263458 -2.401184208 -1.487630879 -1.368579826
##  [76] -0.989775631  1.356114931  1.709392162 -0.633670165  1.770677990
##  [81]  0.274219021  1.000219974 -0.808515059 -0.871916643 -2.151063102
##  [86] -0.350407053 -3.234663999 -2.279539723 -1.114053857 -2.319244900
##  [91] -1.430136190 -2.322809507 -1.975066728 -2.531479937 -2.399113628
##  [96] -0.538788396 -2.628538819 -1.931605552 -1.159526399 -1.014676079
## [101] -0.432185537 -0.549558850  1.264051868  1.430515867  2.598609523
## [106]  0.061890650 -0.617448193  0.276831567  0.101251347 -0.620719712
## [111] -0.443733826  0.087912564  0.726037551 -0.684766508 -1.100115503
## [116] -0.591801408 -1.364273590 -1.231732070  0.123787254 -0.817652540
## [121]  0.574641597 -1.117586460  0.410843111  2.937392529  0.659205141
## [126]  0.941254795  0.681051840  1.142075967  1.496952499  0.558230240
## [131]  0.639936785  1.367521449  1.786498994  1.017829854 -0.003569923
## [136]  0.746169264  0.679962460  1.432293504  1.069743017 -0.230637043
## [141] -0.706067505 -0.275492766 -1.710354850 -0.992195806 -0.425040791
## [146] -0.170944748  1.336619651 -0.579629920 -1.605344237  0.580025068
## [151]  1.487228085  2.614698261  0.206662565  0.165110137 -0.713557580
## [156] -0.983434777 -0.693691169  0.619494728  0.276063826 -0.483104007
## [161]  0.251552914 -0.137485807  0.896558099 -1.131402124  0.558283920
## [166]  1.678193384  2.069400658  0.888396815 -0.415100789 -0.660815187
## [171]  0.748117966 -1.099365355  0.377660091 -0.874150612 -0.606350362
## [176] -0.211369725 -0.847276755  0.468932148  0.020681103  1.499110867
## [181]  3.126409313  0.195770249  2.117325973  1.817108123  1.150626089
## [186]  0.115911690  2.512665732  1.553432269  3.338992399  1.289444828
## [191]  2.029379968  3.382984512  2.457255539  1.684852166  0.535726173
## [196]  0.403799194 -0.283028070  0.444986829 -0.433816175  0.447145774
## [201]  0.823288188  0.702182433  0.827674987  0.861233509  1.451744630
## [206]  0.349452445  0.850459079  0.327858831 -0.702450476 -1.248961917
## [211]  0.008528714  1.104098600 -1.238293635  0.328369561  0.161403931
## [216]  1.205972092  2.472945399  0.328072560  2.255412751  2.861535473
## [221]  1.438188479  1.252317156 -0.858881100 -0.016332851 -2.350702233
## [226] -1.059086108 -1.183245283 -0.729447937 -0.573160493 -0.584550104
## [231] -2.000657480 -2.597426055 -1.566350686 -0.911930377 -0.154070741
## [236]  0.761900065  0.611286982 -1.080733526 -0.934623033 -1.229103342
## [241] -1.519320967  1.469792754 -0.220051190  2.156993581  0.787710783
## [246]  0.837655525  2.535764561  1.600514779  0.836188700  1.122573472
## [251]  0.581180746  0.890620773  0.516259386  1.247959408  1.730458037
## [256]  2.313668714 -0.659434350 -1.363925887 -0.804814278  0.369415933
## [261] -0.049849610 -1.345918269 -0.945586014 -1.011022981 -0.637770952
## [266] -0.106931692 -1.447537800 -1.883412836 -1.528475651  0.067848989
## [271] -0.362758876 -0.281027342 -0.899903984 -0.837891287 -0.486910734
## [276] -0.172382511 -0.743789022 -2.054865504 -1.950659938 -3.411407367
## [281] -3.223933375 -2.625967645 -2.060706822 -1.556568946 -1.930909946
## [286] -3.625011258 -2.280016976  1.036139594 -1.106346059 -1.261263207
## [291]  1.334929741 -0.665011734  0.315274152  0.125254497 -0.676028950
## [296] -1.911686502 -3.376249052
#Yt-1
yt_1_ar2 <- ar2[3:299]
yt_1_ar2
##   [1]  0.929297848  0.705141640  0.249970706  1.630575045  2.804006969
##   [6] -0.632864527 -0.349744995 -0.845998100 -0.688856343 -1.343701238
##  [11] -2.245002413 -1.467312487 -0.456788733 -1.309590275 -1.564831822
##  [16] -0.839692440 -1.685225610 -1.976351619 -1.240349843 -0.300580252
##  [21] -1.364600891 -0.280115127 -1.099028358  0.133222967 -0.472015853
##  [26] -1.261331210 -1.010989626 -2.595626463 -2.539914335  0.135085568
##  [31] -0.587623429  0.062873138 -1.121668605 -0.196751533 -0.598041725
##  [36] -0.725739020 -1.053296316 -1.663004703 -1.422636980 -0.629945133
##  [41] -0.686128345  0.217580950 -0.722747914 -0.160691574  0.101011878
##  [46] -0.566744798  0.073656410  1.243158084  2.470800226  0.711268152
##  [51] -0.290944268 -0.374751162 -0.221706023 -0.364932764  0.703316452
##  [56]  1.867772149  0.955796044  2.469579745  2.993735248  2.635760644
##  [61]  2.769842614  1.551591476  1.846947073 -0.744491076  0.559766909
##  [66] -0.306104106 -0.806377951 -0.469489115  0.150443138 -0.332375194
##  [71] -0.888023314  0.535932096 -0.318263458 -2.401184208 -1.487630879
##  [76] -1.368579826 -0.989775631  1.356114931  1.709392162 -0.633670165
##  [81]  1.770677990  0.274219021  1.000219974 -0.808515059 -0.871916643
##  [86] -2.151063102 -0.350407053 -3.234663999 -2.279539723 -1.114053857
##  [91] -2.319244900 -1.430136190 -2.322809507 -1.975066728 -2.531479937
##  [96] -2.399113628 -0.538788396 -2.628538819 -1.931605552 -1.159526399
## [101] -1.014676079 -0.432185537 -0.549558850  1.264051868  1.430515867
## [106]  2.598609523  0.061890650 -0.617448193  0.276831567  0.101251347
## [111] -0.620719712 -0.443733826  0.087912564  0.726037551 -0.684766508
## [116] -1.100115503 -0.591801408 -1.364273590 -1.231732070  0.123787254
## [121] -0.817652540  0.574641597 -1.117586460  0.410843111  2.937392529
## [126]  0.659205141  0.941254795  0.681051840  1.142075967  1.496952499
## [131]  0.558230240  0.639936785  1.367521449  1.786498994  1.017829854
## [136] -0.003569923  0.746169264  0.679962460  1.432293504  1.069743017
## [141] -0.230637043 -0.706067505 -0.275492766 -1.710354850 -0.992195806
## [146] -0.425040791 -0.170944748  1.336619651 -0.579629920 -1.605344237
## [151]  0.580025068  1.487228085  2.614698261  0.206662565  0.165110137
## [156] -0.713557580 -0.983434777 -0.693691169  0.619494728  0.276063826
## [161] -0.483104007  0.251552914 -0.137485807  0.896558099 -1.131402124
## [166]  0.558283920  1.678193384  2.069400658  0.888396815 -0.415100789
## [171] -0.660815187  0.748117966 -1.099365355  0.377660091 -0.874150612
## [176] -0.606350362 -0.211369725 -0.847276755  0.468932148  0.020681103
## [181]  1.499110867  3.126409313  0.195770249  2.117325973  1.817108123
## [186]  1.150626089  0.115911690  2.512665732  1.553432269  3.338992399
## [191]  1.289444828  2.029379968  3.382984512  2.457255539  1.684852166
## [196]  0.535726173  0.403799194 -0.283028070  0.444986829 -0.433816175
## [201]  0.447145774  0.823288188  0.702182433  0.827674987  0.861233509
## [206]  1.451744630  0.349452445  0.850459079  0.327858831 -0.702450476
## [211] -1.248961917  0.008528714  1.104098600 -1.238293635  0.328369561
## [216]  0.161403931  1.205972092  2.472945399  0.328072560  2.255412751
## [221]  2.861535473  1.438188479  1.252317156 -0.858881100 -0.016332851
## [226] -2.350702233 -1.059086108 -1.183245283 -0.729447937 -0.573160493
## [231] -0.584550104 -2.000657480 -2.597426055 -1.566350686 -0.911930377
## [236] -0.154070741  0.761900065  0.611286982 -1.080733526 -0.934623033
## [241] -1.229103342 -1.519320967  1.469792754 -0.220051190  2.156993581
## [246]  0.787710783  0.837655525  2.535764561  1.600514779  0.836188700
## [251]  1.122573472  0.581180746  0.890620773  0.516259386  1.247959408
## [256]  1.730458037  2.313668714 -0.659434350 -1.363925887 -0.804814278
## [261]  0.369415933 -0.049849610 -1.345918269 -0.945586014 -1.011022981
## [266] -0.637770952 -0.106931692 -1.447537800 -1.883412836 -1.528475651
## [271]  0.067848989 -0.362758876 -0.281027342 -0.899903984 -0.837891287
## [276] -0.486910734 -0.172382511 -0.743789022 -2.054865504 -1.950659938
## [281] -3.411407367 -3.223933375 -2.625967645 -2.060706822 -1.556568946
## [286] -1.930909946 -3.625011258 -2.280016976  1.036139594 -1.106346059
## [291] -1.261263207  1.334929741 -0.665011734  0.315274152  0.125254497
## [296] -0.676028950 -1.911686502
plot(y=yt_ar2,x=yt_1_ar2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).

Yt dengan Yt-2
#Yt-2
yt_2_ar2 <- ar2[2:298]
yt_2_ar2
##   [1] -0.897909868  0.929297848  0.705141640  0.249970706  1.630575045
##   [6]  2.804006969 -0.632864527 -0.349744995 -0.845998100 -0.688856343
##  [11] -1.343701238 -2.245002413 -1.467312487 -0.456788733 -1.309590275
##  [16] -1.564831822 -0.839692440 -1.685225610 -1.976351619 -1.240349843
##  [21] -0.300580252 -1.364600891 -0.280115127 -1.099028358  0.133222967
##  [26] -0.472015853 -1.261331210 -1.010989626 -2.595626463 -2.539914335
##  [31]  0.135085568 -0.587623429  0.062873138 -1.121668605 -0.196751533
##  [36] -0.598041725 -0.725739020 -1.053296316 -1.663004703 -1.422636980
##  [41] -0.629945133 -0.686128345  0.217580950 -0.722747914 -0.160691574
##  [46]  0.101011878 -0.566744798  0.073656410  1.243158084  2.470800226
##  [51]  0.711268152 -0.290944268 -0.374751162 -0.221706023 -0.364932764
##  [56]  0.703316452  1.867772149  0.955796044  2.469579745  2.993735248
##  [61]  2.635760644  2.769842614  1.551591476  1.846947073 -0.744491076
##  [66]  0.559766909 -0.306104106 -0.806377951 -0.469489115  0.150443138
##  [71] -0.332375194 -0.888023314  0.535932096 -0.318263458 -2.401184208
##  [76] -1.487630879 -1.368579826 -0.989775631  1.356114931  1.709392162
##  [81] -0.633670165  1.770677990  0.274219021  1.000219974 -0.808515059
##  [86] -0.871916643 -2.151063102 -0.350407053 -3.234663999 -2.279539723
##  [91] -1.114053857 -2.319244900 -1.430136190 -2.322809507 -1.975066728
##  [96] -2.531479937 -2.399113628 -0.538788396 -2.628538819 -1.931605552
## [101] -1.159526399 -1.014676079 -0.432185537 -0.549558850  1.264051868
## [106]  1.430515867  2.598609523  0.061890650 -0.617448193  0.276831567
## [111]  0.101251347 -0.620719712 -0.443733826  0.087912564  0.726037551
## [116] -0.684766508 -1.100115503 -0.591801408 -1.364273590 -1.231732070
## [121]  0.123787254 -0.817652540  0.574641597 -1.117586460  0.410843111
## [126]  2.937392529  0.659205141  0.941254795  0.681051840  1.142075967
## [131]  1.496952499  0.558230240  0.639936785  1.367521449  1.786498994
## [136]  1.017829854 -0.003569923  0.746169264  0.679962460  1.432293504
## [141]  1.069743017 -0.230637043 -0.706067505 -0.275492766 -1.710354850
## [146] -0.992195806 -0.425040791 -0.170944748  1.336619651 -0.579629920
## [151] -1.605344237  0.580025068  1.487228085  2.614698261  0.206662565
## [156]  0.165110137 -0.713557580 -0.983434777 -0.693691169  0.619494728
## [161]  0.276063826 -0.483104007  0.251552914 -0.137485807  0.896558099
## [166] -1.131402124  0.558283920  1.678193384  2.069400658  0.888396815
## [171] -0.415100789 -0.660815187  0.748117966 -1.099365355  0.377660091
## [176] -0.874150612 -0.606350362 -0.211369725 -0.847276755  0.468932148
## [181]  0.020681103  1.499110867  3.126409313  0.195770249  2.117325973
## [186]  1.817108123  1.150626089  0.115911690  2.512665732  1.553432269
## [191]  3.338992399  1.289444828  2.029379968  3.382984512  2.457255539
## [196]  1.684852166  0.535726173  0.403799194 -0.283028070  0.444986829
## [201] -0.433816175  0.447145774  0.823288188  0.702182433  0.827674987
## [206]  0.861233509  1.451744630  0.349452445  0.850459079  0.327858831
## [211] -0.702450476 -1.248961917  0.008528714  1.104098600 -1.238293635
## [216]  0.328369561  0.161403931  1.205972092  2.472945399  0.328072560
## [221]  2.255412751  2.861535473  1.438188479  1.252317156 -0.858881100
## [226] -0.016332851 -2.350702233 -1.059086108 -1.183245283 -0.729447937
## [231] -0.573160493 -0.584550104 -2.000657480 -2.597426055 -1.566350686
## [236] -0.911930377 -0.154070741  0.761900065  0.611286982 -1.080733526
## [241] -0.934623033 -1.229103342 -1.519320967  1.469792754 -0.220051190
## [246]  2.156993581  0.787710783  0.837655525  2.535764561  1.600514779
## [251]  0.836188700  1.122573472  0.581180746  0.890620773  0.516259386
## [256]  1.247959408  1.730458037  2.313668714 -0.659434350 -1.363925887
## [261] -0.804814278  0.369415933 -0.049849610 -1.345918269 -0.945586014
## [266] -1.011022981 -0.637770952 -0.106931692 -1.447537800 -1.883412836
## [271] -1.528475651  0.067848989 -0.362758876 -0.281027342 -0.899903984
## [276] -0.837891287 -0.486910734 -0.172382511 -0.743789022 -2.054865504
## [281] -1.950659938 -3.411407367 -3.223933375 -2.625967645 -2.060706822
## [286] -1.556568946 -1.930909946 -3.625011258 -2.280016976  1.036139594
## [291] -1.106346059 -1.261263207  1.334929741 -0.665011734  0.315274152
## [296]  0.125254497 -0.676028950
plot(y=yt_ar2,x=yt_2_ar2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).

Yt dengan Yt-3
#Yt-3
yt_3_ar2 <- ar2[1:297]
yt_3_ar2
##   [1]  0.131540851 -0.897909868  0.929297848  0.705141640  0.249970706
##   [6]  1.630575045  2.804006969 -0.632864527 -0.349744995 -0.845998100
##  [11] -0.688856343 -1.343701238 -2.245002413 -1.467312487 -0.456788733
##  [16] -1.309590275 -1.564831822 -0.839692440 -1.685225610 -1.976351619
##  [21] -1.240349843 -0.300580252 -1.364600891 -0.280115127 -1.099028358
##  [26]  0.133222967 -0.472015853 -1.261331210 -1.010989626 -2.595626463
##  [31] -2.539914335  0.135085568 -0.587623429  0.062873138 -1.121668605
##  [36] -0.196751533 -0.598041725 -0.725739020 -1.053296316 -1.663004703
##  [41] -1.422636980 -0.629945133 -0.686128345  0.217580950 -0.722747914
##  [46] -0.160691574  0.101011878 -0.566744798  0.073656410  1.243158084
##  [51]  2.470800226  0.711268152 -0.290944268 -0.374751162 -0.221706023
##  [56] -0.364932764  0.703316452  1.867772149  0.955796044  2.469579745
##  [61]  2.993735248  2.635760644  2.769842614  1.551591476  1.846947073
##  [66] -0.744491076  0.559766909 -0.306104106 -0.806377951 -0.469489115
##  [71]  0.150443138 -0.332375194 -0.888023314  0.535932096 -0.318263458
##  [76] -2.401184208 -1.487630879 -1.368579826 -0.989775631  1.356114931
##  [81]  1.709392162 -0.633670165  1.770677990  0.274219021  1.000219974
##  [86] -0.808515059 -0.871916643 -2.151063102 -0.350407053 -3.234663999
##  [91] -2.279539723 -1.114053857 -2.319244900 -1.430136190 -2.322809507
##  [96] -1.975066728 -2.531479937 -2.399113628 -0.538788396 -2.628538819
## [101] -1.931605552 -1.159526399 -1.014676079 -0.432185537 -0.549558850
## [106]  1.264051868  1.430515867  2.598609523  0.061890650 -0.617448193
## [111]  0.276831567  0.101251347 -0.620719712 -0.443733826  0.087912564
## [116]  0.726037551 -0.684766508 -1.100115503 -0.591801408 -1.364273590
## [121] -1.231732070  0.123787254 -0.817652540  0.574641597 -1.117586460
## [126]  0.410843111  2.937392529  0.659205141  0.941254795  0.681051840
## [131]  1.142075967  1.496952499  0.558230240  0.639936785  1.367521449
## [136]  1.786498994  1.017829854 -0.003569923  0.746169264  0.679962460
## [141]  1.432293504  1.069743017 -0.230637043 -0.706067505 -0.275492766
## [146] -1.710354850 -0.992195806 -0.425040791 -0.170944748  1.336619651
## [151] -0.579629920 -1.605344237  0.580025068  1.487228085  2.614698261
## [156]  0.206662565  0.165110137 -0.713557580 -0.983434777 -0.693691169
## [161]  0.619494728  0.276063826 -0.483104007  0.251552914 -0.137485807
## [166]  0.896558099 -1.131402124  0.558283920  1.678193384  2.069400658
## [171]  0.888396815 -0.415100789 -0.660815187  0.748117966 -1.099365355
## [176]  0.377660091 -0.874150612 -0.606350362 -0.211369725 -0.847276755
## [181]  0.468932148  0.020681103  1.499110867  3.126409313  0.195770249
## [186]  2.117325973  1.817108123  1.150626089  0.115911690  2.512665732
## [191]  1.553432269  3.338992399  1.289444828  2.029379968  3.382984512
## [196]  2.457255539  1.684852166  0.535726173  0.403799194 -0.283028070
## [201]  0.444986829 -0.433816175  0.447145774  0.823288188  0.702182433
## [206]  0.827674987  0.861233509  1.451744630  0.349452445  0.850459079
## [211]  0.327858831 -0.702450476 -1.248961917  0.008528714  1.104098600
## [216] -1.238293635  0.328369561  0.161403931  1.205972092  2.472945399
## [221]  0.328072560  2.255412751  2.861535473  1.438188479  1.252317156
## [226] -0.858881100 -0.016332851 -2.350702233 -1.059086108 -1.183245283
## [231] -0.729447937 -0.573160493 -0.584550104 -2.000657480 -2.597426055
## [236] -1.566350686 -0.911930377 -0.154070741  0.761900065  0.611286982
## [241] -1.080733526 -0.934623033 -1.229103342 -1.519320967  1.469792754
## [246] -0.220051190  2.156993581  0.787710783  0.837655525  2.535764561
## [251]  1.600514779  0.836188700  1.122573472  0.581180746  0.890620773
## [256]  0.516259386  1.247959408  1.730458037  2.313668714 -0.659434350
## [261] -1.363925887 -0.804814278  0.369415933 -0.049849610 -1.345918269
## [266] -0.945586014 -1.011022981 -0.637770952 -0.106931692 -1.447537800
## [271] -1.883412836 -1.528475651  0.067848989 -0.362758876 -0.281027342
## [276] -0.899903984 -0.837891287 -0.486910734 -0.172382511 -0.743789022
## [281] -2.054865504 -1.950659938 -3.411407367 -3.223933375 -2.625967645
## [286] -2.060706822 -1.556568946 -1.930909946 -3.625011258 -2.280016976
## [291]  1.036139594 -1.106346059 -1.261263207  1.334929741 -0.665011734
## [296]  0.315274152  0.125254497
plot(y=yt_ar2,x=yt_3_ar2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-3}\).

Korelasi Antar Lag

Data Bangkitan
cor(yt_ar2, yt_1_ar2, use = "complete.obs")
## [1] 0.6334386

Nilai korelasi antara \(Y_t\) dan \(Y_{t-1}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:

\[ \rho_{1} = \frac{\phi_1}{1-\phi_2} = \frac{0.5}{1 - 0.2} = 0.625 \]

cor(yt_ar2, yt_2_ar2, use = "complete.obs")
## [1] 0.513792

Nilai korelasi antara \(Y_t\) dan \(Y_{t-2}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:

\[ \rho_{2} = \frac{\phi_2 (1-\phi_2) + (\phi_1)^2}{1-\phi_2} = \frac{0.2(1-0.2) + (0.5)^2}{1 - 0.2} \approx 0.512 \]

cor(yt_ar2, yt_3_ar2, use = "complete.obs")
## [1] 0.4262701

Nilai korelasi antara \(Y_t\) dan \(Y_{t-3}\) dari hasil simulasi mendekati perhitungan teoritis yaitu:

\[ \rho_{3} = \phi_1\rho_{3-1} + \phi_2\rho_{3-2} = \phi_1\rho_{2} + \phi_2\rho_{1} = 0.5(0.512) + 0.2(0.625) \approx 0.381 \]

Kesimpulan: Berdasarkan hasil analisis, model AR(2) merupakan model yang paling sesuai karena pola ACF dan PACF sudah konsisten dengan karakteristik teoritis AR(2), serta nilai autokorelasi empiris mendekati nilai teoritis. Meskipun hasil EACF menunjukkan kecenderungan ke ARMA(1,1), hal tersebut kemungkinan dipengaruhi oleh keterbatasan sampel atau kemiripan pola antar model. Dengan demikian, AR(2) tetap dapat dianggap sebagai representasi yang tepat untuk proses data bangkitan.

Proses ARMA(2,2)

Pembangkitan Proses ARMA(2,2)

Rumus ARMA (2,2): \[ Y_t = \phi_1Y_{t-1}+\phi_2Y_{t-2}+e_t-\theta_1e_{t-1}-\theta_2e_{t-2} \] dengan \(\theta_1\) = 0.4, \(\theta_2\) = 0.6, \(\phi_1\) = 0.5, dan \(\phi_2\) = 0.2.

# Manual
set.seed(1010)

y.arma <- wn[1:2]   

for (i in 3:n) {
  y.arma[i] <- (0.5 * y.arma[i-1]) + 
               (0.2 * y.arma[i-2]) +
               wn[i] +
               (0.4 * wn[i-1]) +
               (0.6 * wn[i-2])
}
y.arma
##   [1]  0.1315408507 -0.8979098678  0.6490584112  0.5670538453  1.1040755646
##   [6]  2.1666708551  3.6156245817  1.4743903819  1.0850479575 -1.3613861046
##  [11] -1.2338813084 -2.1243862581 -3.1939242710 -3.1701066986 -2.3896269389
##  [16] -2.3718636432 -2.3621087157 -2.2508971826 -2.9596341124 -3.1539771138
##  [21] -3.0418122366 -1.9823683078 -2.2289187469 -1.0062089885 -2.0297627901
##  [26] -0.4744024462 -1.0781017468 -1.3701718025 -1.7987072509 -3.7568024608
##  [31] -4.1847445327 -2.4382452466 -2.0575295717 -0.0911186175 -1.4490886232
##  [36] -0.6076914446 -1.3497407211 -1.0830045097 -1.7024153434 -2.5197654093
##  [41] -2.7198157113 -2.1968020303 -1.7916880403 -0.4348370521 -1.0473922237
##  [46] -0.3192419282 -0.3969133154 -0.6227548509 -0.0924342754  0.9325738506
##  [51]  3.0122573682  2.4454831403  1.4760431644 -0.0643679508 -0.5461730279
##  [56] -0.6784658549  0.4243197444  1.9301390808  2.1248947815  3.9725614580
##  [61]  4.5550447761  5.3150025930  5.6203880226  4.2409849098  4.1294892331
##  [66]  0.9252426396  1.3701387231 -0.5288919875 -0.5929594477 -0.9757027592
##  [71] -0.5211792791 -0.5538914081 -0.9307075093 -0.0187023456 -0.6367046077
##  [76] -2.2069303334 -2.6390626365 -3.4043427022 -2.4297860883  0.1390567834
##  [81]  1.6579727565  0.8637556592  2.5428452214  0.6022881181  2.1723143762
##  [86] -0.2438956572 -0.5951906828 -2.9849387954 -1.7339822802 -4.6654646819
##  [91] -3.7836495546 -3.9666681459 -4.1325902764 -3.0262664639 -4.2864109226
##  [96] -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097 -4.2835223541
## [101] -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084 -1.3312387130
## [106]  0.7849170050  1.6064013034  3.9292469905  1.9596439795  0.9664737813
## [111]  0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023 -0.4620127937
## [116]  0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147 -2.2610634558
## [121] -2.1325223511 -1.1874697279 -1.5071768803  0.3218529332 -1.3783213458
## [126]  0.3085934849  2.4311778976  2.0806680196  2.9673723690  1.4530768428
## [131]  1.9792495800  2.3624139896  1.8422568201  1.7614003802  1.9584343073
## [136]  2.7174696443  2.5529423211  1.4754614153  1.3554392078  0.9762882119
## [141]  2.1519800469  2.0506378942  1.0566362664 -0.1564765121 -0.6963019933
## [146] -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485  1.0132172775
## [151] -0.1475489089 -1.0352244145 -0.4098905792  0.7560315697  3.5576045354
## [156]  2.1448787201  1.8165941197 -0.5235159857 -1.1697917262 -1.5151996272
## [161] -0.2480426055  0.1076470164 -0.0009816395  0.2239496072 -0.3267270455
## [166]  0.9924955244 -0.8552703690  0.6436579300  1.2226656780  3.0756483638
## [171]  2.7230731085  1.1818983317 -0.2938174133  0.2347314183 -1.1966072807
## [176]  0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367 -1.2956348622
## [181]  0.0031996111 -0.3001120910  1.7887425967  3.7384623212  2.3458004938
## [186]  4.0714796600  2.7815006616  3.1478649220  1.6664269995  3.2494060618
## [191]  2.6280455760  5.4679647463  3.5571011493  4.5485533388  4.9684033961
## [196]  5.0280773241  4.6975450886  2.6840203628  1.6290009627  0.1999273113
## [201]  0.5740551176 -0.4256382849  0.5406114019  0.7418567933  1.2997851726
## [206]  1.6025208730  1.6136129632  2.2928430253  1.4468904019  1.8612868351
## [211]  0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389  0.3581329351
## [216] -0.7915369668  0.4955112665 -0.4502244253  1.4675554008  3.0521765941
## [221]  2.0408339742  3.8704090143  3.9605441093  3.9360503187  3.5445138311
## [226]  0.5049588491  0.3915050022 -2.8725640338 -2.0091667120 -3.0173010664
## [231] -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178 -3.7484191100
## [236] -3.8057155958 -3.0969262848 -1.4586533030  0.1531135428  0.8236045641
## [241] -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239  0.1246023618
## [246] -0.5437266688  2.9508487574  1.5184775014  2.4469359872  3.3434532407
## [251]  3.1174139183  2.9978533478  2.4173578196  1.5319233546  1.7966371548
## [256]  1.2212161425  1.9888356266  2.5393974322  3.7546275738  1.3043079574
## [261] -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038 -1.1442085530
## [266] -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607 -1.8729730474
## [271] -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715 -0.3854214991
## [276] -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763 -1.1048884660
## [281] -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841 -5.9623854154
## [286] -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036 -4.8885674466
## [291] -2.0508739512 -2.0599004072 -1.0821178750  0.1666168226 -0.8877977615
## [296]  0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839 -4.5465410226
# Dengan fungsi arima.sim
arma22 <- arima.sim(
  list(order=c(2,0,2),
       ar = c(0.5, 0.2),   
       ma = c(0.4, 0.6)),  
  n = 300
)
arma22
## Time Series:
## Start = 1 
## End = 300 
## Frequency = 1 
##   [1] -1.3696174977 -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230
##   [6] -2.0573868423 -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124
##  [11] -1.0829677579 -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169
##  [16] -1.7916785770 -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190
##  [21] -0.6227524142 -0.0924324177  0.9325752668  3.0122584478  2.4454839633
##  [26]  1.4760437919 -0.0643674725 -0.5461726632 -0.6784655769  0.4243199563
##  [31]  1.9301392423  2.1248949046  3.9725615519  4.5550448477  5.3150026476
##  [36]  5.6203880642  4.2409849415  4.1294892573  0.9252426580  1.3701387371
##  [41] -0.5288919768 -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045
##  [46] -0.9307075065 -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356
##  [51] -3.4043427015 -2.4297860877  0.1390567838  1.6579727568  0.8637556594
##  [56]  2.5428452215  0.6022881183  2.1723143763 -0.2438956571 -0.5951906827
##  [61] -2.9849387954 -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459
##  [66] -4.1325902763 -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322
##  [71] -4.5967456399 -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106
##  [76] -2.6374499698 -1.5337718084 -1.3312387130  0.7849170050  1.6064013034
##  [81]  3.9292469905  1.9596439795  0.9664737813  0.0669866797 -0.1584849418
##  [86] -0.4141202333 -0.6312709023 -0.4620127937  0.4949622808 -0.3416039498
##  [91] -0.9383995764 -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279
##  [96] -1.5071768803  0.3218529332 -1.3783213458  0.3085934849  2.4311778976
## [101]  2.0806680196  2.9673723690  1.4530768428  1.9792495800  2.3624139896
## [106]  1.8422568201  1.7614003802  1.9584343073  2.7174696443  2.5529423211
## [111]  1.4754614153  1.3554392078  0.9762882119  2.1519800469  2.0506378942
## [116]  1.0566362664 -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056
## [121] -1.8481320229 -0.9362785485  1.0132172775 -0.1475489089 -1.0352244145
## [126] -0.4098905792  0.7560315697  3.5576045354  2.1448787201  1.8165941197
## [131] -0.5235159857 -1.1697917262 -1.5151996272 -0.2480426055  0.1076470164
## [136] -0.0009816395  0.2239496072 -0.3267270455  0.9924955244 -0.8552703690
## [141]  0.6436579300  1.2226656780  3.0756483638  2.7230731085  1.1818983317
## [146] -0.2938174133  0.2347314183 -1.1966072807  0.3867847283 -1.3827057885
## [151] -0.7294145525 -0.9784002367 -1.2956348622  0.0031996111 -0.3001120910
## [156]  1.7887425967  3.7384623212  2.3458004938  4.0714796600  2.7815006616
## [161]  3.1478649220  1.6664269995  3.2494060618  2.6280455760  5.4679647463
## [166]  3.5571011493  4.5485533388  4.9684033961  5.0280773241  4.6975450886
## [171]  2.6840203628  1.6290009627  0.1999273113  0.5740551176 -0.4256382849
## [176]  0.5406114019  0.7418567933  1.2997851726  1.6025208730  1.6136129632
## [181]  2.2928430253  1.4468904019  1.8612868351  0.8777139291 -0.0610314965
## [186] -1.3332268089 -0.9125263389  0.3581329351 -0.7915369668  0.4955112665
## [191] -0.4502244253  1.4675554008  3.0521765941  2.0408339742  3.8704090143
## [196]  3.9605441093  3.9360503187  3.5445138311  0.5049588491  0.3915050022
## [201] -2.8725640338 -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384
## [206] -1.2514830640 -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848
## [211] -1.4586533030  0.1531135428  0.8236045641 -0.3790786936 -1.0001442542
## [216] -2.2513926710 -2.5717361239  0.1246023618 -0.5437266688  2.9508487574
## [221]  1.5184775014  2.4469359872  3.3434532407  3.1174139183  2.9978533478
## [226]  2.4173578196  1.5319233546  1.7966371548  1.2212161425  1.9888356266
## [231]  2.5393974322  3.7546275738  1.3043079574 -0.2394983994 -1.7460452432
## [236] -0.7708653105 -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475
## [241] -1.6095317523 -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658
## [246] -1.6735889734 -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859
## [251] -1.3620096392 -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522
## [256] -5.4245906442 -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618
## [261] -3.7899616180 -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072
## [266] -1.0821178750  0.1666168226 -0.8877977615  0.8502273032 -0.1476428826
## [271] -0.4367626603 -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874
## [276] -1.9178765546 -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804
## [281]  0.0122011943  1.4282807605  0.6560697941  0.7567608782 -0.8228605730
## [286] -1.1860435575 -1.8041134735 -0.4466562647 -0.3954967938  2.1487643901
## [291]  0.9586140341  2.2653125680  0.2588626762  1.0329576284 -0.9297517508
## [296] -1.2122608974 -2.3845725749 -5.0393956847 -3.3418821354 -4.8054361842

Karakteristik ARMA(2,2)

Plot Time Series
par(mfrow = c(1, 2))
ts.plot(y.arma)
ts.plot(arma22)

Berdasarkan plot time series tersebut, terlihat bahwa model ARMA(2,2) cenderung stasioner dalam rataan.

Plot ACF
par(mfrow = c(1, 2))
acf(y.arma)
acf(arma22)

Berdasarkan plot ACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot ACF yang tails off, sesuai dengan teori yang ada.

Plot PACF
par(mfrow = c(1, 2))
pacf(y.arma)
pacf(arma22)

Berdasarkan plot PACF tersebut, terlihat bahwa model ARMA(2,2) hasil simulasi memiliki plot PACF yang tails off, sesuai dengan teori yang ada.

Plot EACF
TSA::eacf(y.arma)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x o  o  o  o 
## 1 o x o o o o o o o o o  o  o  o 
## 2 x x x o o o o o o o o  o  o  o 
## 3 x x o o o o o o o o o  o  o  o 
## 4 x x x o o o o o o o o  o  o  o 
## 5 x x x o o o o o o o o  o  o  o 
## 6 x x x o o o o o o o o  o  o  o 
## 7 x x x o o o o o o o o  o  o  o
TSA::eacf(arma22)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 x x x x x x x x x x x  x  o  o 
## 1 o x o o o o o o o o o  o  o  o 
## 2 x x x o o o o o o o o  o  o  o 
## 3 x x x o o o o o o o o  o  o  o 
## 4 x x o o o o o o o o o  o  o  o 
## 5 x x x o x o o o o o o  o  o  o 
## 6 o x o o x o o o o o o  o  o  o 
## 7 x x x o x x o o o o o  o  o  o

Berdasarkan pola segitiga nol pada plot EACF, terlihat bahwa segitiga nol berada pada ordo AR(1) dan ordo MA(2).

Scatterplot Antar Lag

Yt dengan Yt-1
#Yt
yt_arma2 <- arma22[4:300]   
yt_arma2
##   [1] -4.1844989443 -2.4380580230 -2.0573868423 -0.0910098080 -1.4490056726
##   [6] -0.6076282073 -1.3496925124 -1.0829677579 -1.7023873257 -2.5197440501
##  [11] -2.7197994281 -2.1967896169 -1.7916785770 -0.4348298378 -1.0473867239
##  [16] -0.3192377354 -0.3969101190 -0.6227524142 -0.0924324177  0.9325752668
##  [21]  3.0122584478  2.4454839633  1.4760437919 -0.0643674725 -0.5461726632
##  [26] -0.6784655769  0.4243199563  1.9301392423  2.1248949046  3.9725615519
##  [31]  4.5550448477  5.3150026476  5.6203880642  4.2409849415  4.1294892573
##  [36]  0.9252426580  1.3701387371 -0.5288919768 -0.5929594395 -0.9757027529
##  [41] -0.5211792743 -0.5538914045 -0.9307075065 -0.0187023435 -0.6367046061
##  [46] -2.2069303322 -2.6390626356 -3.4043427015 -2.4297860877  0.1390567838
##  [51]  1.6579727568  0.8637556594  2.5428452215  0.6022881183  2.1723143763
##  [56] -0.2438956571 -0.5951906827 -2.9849387954 -1.7339822802 -4.6654646818
##  [61] -3.7836495546 -3.9666681459 -4.1325902763 -3.0262664639 -4.2864109226
##  [66] -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097 -4.2835223541
##  [71] -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084 -1.3312387130
##  [76]  0.7849170050  1.6064013034  3.9292469905  1.9596439795  0.9664737813
##  [81]  0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023 -0.4620127937
##  [86]  0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147 -2.2610634558
##  [91] -2.1325223511 -1.1874697279 -1.5071768803  0.3218529332 -1.3783213458
##  [96]  0.3085934849  2.4311778976  2.0806680196  2.9673723690  1.4530768428
## [101]  1.9792495800  2.3624139896  1.8422568201  1.7614003802  1.9584343073
## [106]  2.7174696443  2.5529423211  1.4754614153  1.3554392078  0.9762882119
## [111]  2.1519800469  2.0506378942  1.0566362664 -0.1564765121 -0.6963019933
## [116] -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485  1.0132172775
## [121] -0.1475489089 -1.0352244145 -0.4098905792  0.7560315697  3.5576045354
## [126]  2.1448787201  1.8165941197 -0.5235159857 -1.1697917262 -1.5151996272
## [131] -0.2480426055  0.1076470164 -0.0009816395  0.2239496072 -0.3267270455
## [136]  0.9924955244 -0.8552703690  0.6436579300  1.2226656780  3.0756483638
## [141]  2.7230731085  1.1818983317 -0.2938174133  0.2347314183 -1.1966072807
## [146]  0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367 -1.2956348622
## [151]  0.0031996111 -0.3001120910  1.7887425967  3.7384623212  2.3458004938
## [156]  4.0714796600  2.7815006616  3.1478649220  1.6664269995  3.2494060618
## [161]  2.6280455760  5.4679647463  3.5571011493  4.5485533388  4.9684033961
## [166]  5.0280773241  4.6975450886  2.6840203628  1.6290009627  0.1999273113
## [171]  0.5740551176 -0.4256382849  0.5406114019  0.7418567933  1.2997851726
## [176]  1.6025208730  1.6136129632  2.2928430253  1.4468904019  1.8612868351
## [181]  0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389  0.3581329351
## [186] -0.7915369668  0.4955112665 -0.4502244253  1.4675554008  3.0521765941
## [191]  2.0408339742  3.8704090143  3.9605441093  3.9360503187  3.5445138311
## [196]  0.5049588491  0.3915050022 -2.8725640338 -2.0091667120 -3.0173010664
## [201] -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178 -3.7484191100
## [206] -3.8057155958 -3.0969262848 -1.4586533030  0.1531135428  0.8236045641
## [211] -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239  0.1246023618
## [216] -0.5437266688  2.9508487574  1.5184775014  2.4469359872  3.3434532407
## [221]  3.1174139183  2.9978533478  2.4173578196  1.5319233546  1.7966371548
## [226]  1.2212161425  1.9888356266  2.5393974322  3.7546275738  1.3043079574
## [231] -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038 -1.1442085530
## [236] -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607 -1.8729730474
## [241] -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715 -0.3854214991
## [246] -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763 -1.1048884660
## [251] -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841 -5.9623854154
## [256] -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036 -4.8885674466
## [261] -2.0508739512 -2.0599004072 -1.0821178750  0.1666168226 -0.8877977615
## [266]  0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839 -4.5465410226
## [271] -4.8087456252 -4.2289076874 -1.9178765546 -2.2804864851 -0.4098672855
## [276] -1.6178574433 -0.0137269804  0.0122011943  1.4282807605  0.6560697941
## [281]  0.7567608782 -0.8228605730 -1.1860435575 -1.8041134735 -0.4466562647
## [286] -0.3954967938  2.1487643901  0.9586140341  2.2653125680  0.2588626762
## [291]  1.0329576284 -0.9297517508 -1.2122608974 -2.3845725749 -5.0393956847
## [296] -3.3418821354 -4.8054361842
#Yt-1
yt_1_arma2 <- arma22[3:299]
yt_1_arma2
##   [1] -3.7564803134 -4.1844989443 -2.4380580230 -2.0573868423 -0.0910098080
##   [6] -1.4490056726 -0.6076282073 -1.3496925124 -1.0829677579 -1.7023873257
##  [11] -2.5197440501 -2.7197994281 -2.1967896169 -1.7916785770 -0.4348298378
##  [16] -1.0473867239 -0.3192377354 -0.3969101190 -0.6227524142 -0.0924324177
##  [21]  0.9325752668  3.0122584478  2.4454839633  1.4760437919 -0.0643674725
##  [26] -0.5461726632 -0.6784655769  0.4243199563  1.9301392423  2.1248949046
##  [31]  3.9725615519  4.5550448477  5.3150026476  5.6203880642  4.2409849415
##  [36]  4.1294892573  0.9252426580  1.3701387371 -0.5288919768 -0.5929594395
##  [41] -0.9757027529 -0.5211792743 -0.5538914045 -0.9307075065 -0.0187023435
##  [46] -0.6367046061 -2.2069303322 -2.6390626356 -3.4043427015 -2.4297860877
##  [51]  0.1390567838  1.6579727568  0.8637556594  2.5428452215  0.6022881183
##  [56]  2.1723143763 -0.2438956571 -0.5951906827 -2.9849387954 -1.7339822802
##  [61] -4.6654646818 -3.7836495546 -3.9666681459 -4.1325902763 -3.0262664639
##  [66] -4.2864109226 -3.7622722444 -4.7151923322 -4.5967456399 -3.0173218097
##  [71] -4.2835223541 -3.3062941169 -3.5092919106 -2.6374499698 -1.5337718084
##  [76] -1.3312387130  0.7849170050  1.6064013034  3.9292469905  1.9596439795
##  [81]  0.9664737813  0.0669866797 -0.1584849418 -0.4141202333 -0.6312709023
##  [86] -0.4620127937  0.4949622808 -0.3416039498 -0.9383995764 -1.4427075147
##  [91] -2.2610634558 -2.1325223511 -1.1874697279 -1.5071768803  0.3218529332
##  [96] -1.3783213458  0.3085934849  2.4311778976  2.0806680196  2.9673723690
## [101]  1.4530768428  1.9792495800  2.3624139896  1.8422568201  1.7614003802
## [106]  1.9584343073  2.7174696443  2.5529423211  1.4754614153  1.3554392078
## [111]  0.9762882119  2.1519800469  2.0506378942  1.0566362664 -0.1564765121
## [116] -0.6963019933 -2.2441924588 -1.8416334056 -1.8481320229 -0.9362785485
## [121]  1.0132172775 -0.1475489089 -1.0352244145 -0.4098905792  0.7560315697
## [126]  3.5576045354  2.1448787201  1.8165941197 -0.5235159857 -1.1697917262
## [131] -1.5151996272 -0.2480426055  0.1076470164 -0.0009816395  0.2239496072
## [136] -0.3267270455  0.9924955244 -0.8552703690  0.6436579300  1.2226656780
## [141]  3.0756483638  2.7230731085  1.1818983317 -0.2938174133  0.2347314183
## [146] -1.1966072807  0.3867847283 -1.3827057885 -0.7294145525 -0.9784002367
## [151] -1.2956348622  0.0031996111 -0.3001120910  1.7887425967  3.7384623212
## [156]  2.3458004938  4.0714796600  2.7815006616  3.1478649220  1.6664269995
## [161]  3.2494060618  2.6280455760  5.4679647463  3.5571011493  4.5485533388
## [166]  4.9684033961  5.0280773241  4.6975450886  2.6840203628  1.6290009627
## [171]  0.1999273113  0.5740551176 -0.4256382849  0.5406114019  0.7418567933
## [176]  1.2997851726  1.6025208730  1.6136129632  2.2928430253  1.4468904019
## [181]  1.8612868351  0.8777139291 -0.0610314965 -1.3332268089 -0.9125263389
## [186]  0.3581329351 -0.7915369668  0.4955112665 -0.4502244253  1.4675554008
## [191]  3.0521765941  2.0408339742  3.8704090143  3.9605441093  3.9360503187
## [196]  3.5445138311  0.5049588491  0.3915050022 -2.8725640338 -2.0091667120
## [201] -3.0173010664 -1.8381977156 -1.5748868384 -1.2514830640 -2.5783738178
## [206] -3.7484191100 -3.8057155958 -3.0969262848 -1.4586533030  0.1531135428
## [211]  0.8236045641 -0.3790786936 -1.0001442542 -2.2513926710 -2.5717361239
## [216]  0.1246023618 -0.5437266688  2.9508487574  1.5184775014  2.4469359872
## [221]  3.3434532407  3.1174139183  2.9978533478  2.4173578196  1.5319233546
## [226]  1.7966371548  1.2212161425  1.9888356266  2.5393974322  3.7546275738
## [231]  1.3043079574 -0.2394983994 -1.7460452432 -0.7708653105 -0.3849718038
## [236] -1.1442085530 -1.5138630876 -2.1968083475 -1.6095317523 -0.9686538607
## [241] -1.8729730474 -2.5265869712 -3.1503634658 -1.6735889734 -1.2527046715
## [246] -0.3854214991 -1.2299702469 -1.3664692859 -1.3620096392 -0.8698815763
## [251] -1.1048884660 -2.4558106193 -3.2188795522 -5.4245906442 -5.7588922841
## [256] -5.9623854154 -5.0454539055 -3.9564322618 -3.7899616180 -5.3313166036
## [261] -4.8885674466 -2.0508739512 -2.0599004072 -1.0821178750  0.1666168226
## [266] -0.8877977615  0.8502273032 -0.1476428826 -0.4367626603 -2.1069453839
## [271] -4.5465410226 -4.8087456252 -4.2289076874 -1.9178765546 -2.2804864851
## [276] -0.4098672855 -1.6178574433 -0.0137269804  0.0122011943  1.4282807605
## [281]  0.6560697941  0.7567608782 -0.8228605730 -1.1860435575 -1.8041134735
## [286] -0.4466562647 -0.3954967938  2.1487643901  0.9586140341  2.2653125680
## [291]  0.2588626762  1.0329576284 -0.9297517508 -1.2122608974 -2.3845725749
## [296] -5.0393956847 -3.3418821354
plot(y=yt_arma2,x=yt_1_arma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-1}\).

Yt dengan Yt-2
#Yt-2
yt_2_arma2 <- arma22[2:298]
yt_2_arma2
##   [1] -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230 -2.0573868423
##   [6] -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124 -1.0829677579
##  [11] -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169 -1.7916785770
##  [16] -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190 -0.6227524142
##  [21] -0.0924324177  0.9325752668  3.0122584478  2.4454839633  1.4760437919
##  [26] -0.0643674725 -0.5461726632 -0.6784655769  0.4243199563  1.9301392423
##  [31]  2.1248949046  3.9725615519  4.5550448477  5.3150026476  5.6203880642
##  [36]  4.2409849415  4.1294892573  0.9252426580  1.3701387371 -0.5288919768
##  [41] -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045 -0.9307075065
##  [46] -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356 -3.4043427015
##  [51] -2.4297860877  0.1390567838  1.6579727568  0.8637556594  2.5428452215
##  [56]  0.6022881183  2.1723143763 -0.2438956571 -0.5951906827 -2.9849387954
##  [61] -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459 -4.1325902763
##  [66] -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322 -4.5967456399
##  [71] -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106 -2.6374499698
##  [76] -1.5337718084 -1.3312387130  0.7849170050  1.6064013034  3.9292469905
##  [81]  1.9596439795  0.9664737813  0.0669866797 -0.1584849418 -0.4141202333
##  [86] -0.6312709023 -0.4620127937  0.4949622808 -0.3416039498 -0.9383995764
##  [91] -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279 -1.5071768803
##  [96]  0.3218529332 -1.3783213458  0.3085934849  2.4311778976  2.0806680196
## [101]  2.9673723690  1.4530768428  1.9792495800  2.3624139896  1.8422568201
## [106]  1.7614003802  1.9584343073  2.7174696443  2.5529423211  1.4754614153
## [111]  1.3554392078  0.9762882119  2.1519800469  2.0506378942  1.0566362664
## [116] -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056 -1.8481320229
## [121] -0.9362785485  1.0132172775 -0.1475489089 -1.0352244145 -0.4098905792
## [126]  0.7560315697  3.5576045354  2.1448787201  1.8165941197 -0.5235159857
## [131] -1.1697917262 -1.5151996272 -0.2480426055  0.1076470164 -0.0009816395
## [136]  0.2239496072 -0.3267270455  0.9924955244 -0.8552703690  0.6436579300
## [141]  1.2226656780  3.0756483638  2.7230731085  1.1818983317 -0.2938174133
## [146]  0.2347314183 -1.1966072807  0.3867847283 -1.3827057885 -0.7294145525
## [151] -0.9784002367 -1.2956348622  0.0031996111 -0.3001120910  1.7887425967
## [156]  3.7384623212  2.3458004938  4.0714796600  2.7815006616  3.1478649220
## [161]  1.6664269995  3.2494060618  2.6280455760  5.4679647463  3.5571011493
## [166]  4.5485533388  4.9684033961  5.0280773241  4.6975450886  2.6840203628
## [171]  1.6290009627  0.1999273113  0.5740551176 -0.4256382849  0.5406114019
## [176]  0.7418567933  1.2997851726  1.6025208730  1.6136129632  2.2928430253
## [181]  1.4468904019  1.8612868351  0.8777139291 -0.0610314965 -1.3332268089
## [186] -0.9125263389  0.3581329351 -0.7915369668  0.4955112665 -0.4502244253
## [191]  1.4675554008  3.0521765941  2.0408339742  3.8704090143  3.9605441093
## [196]  3.9360503187  3.5445138311  0.5049588491  0.3915050022 -2.8725640338
## [201] -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384 -1.2514830640
## [206] -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848 -1.4586533030
## [211]  0.1531135428  0.8236045641 -0.3790786936 -1.0001442542 -2.2513926710
## [216] -2.5717361239  0.1246023618 -0.5437266688  2.9508487574  1.5184775014
## [221]  2.4469359872  3.3434532407  3.1174139183  2.9978533478  2.4173578196
## [226]  1.5319233546  1.7966371548  1.2212161425  1.9888356266  2.5393974322
## [231]  3.7546275738  1.3043079574 -0.2394983994 -1.7460452432 -0.7708653105
## [236] -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475 -1.6095317523
## [241] -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658 -1.6735889734
## [246] -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859 -1.3620096392
## [251] -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522 -5.4245906442
## [256] -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618 -3.7899616180
## [261] -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072 -1.0821178750
## [266]  0.1666168226 -0.8877977615  0.8502273032 -0.1476428826 -0.4367626603
## [271] -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874 -1.9178765546
## [276] -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804  0.0122011943
## [281]  1.4282807605  0.6560697941  0.7567608782 -0.8228605730 -1.1860435575
## [286] -1.8041134735 -0.4466562647 -0.3954967938  2.1487643901  0.9586140341
## [291]  2.2653125680  0.2588626762  1.0329576284 -0.9297517508 -1.2122608974
## [296] -2.3845725749 -5.0393956847
plot(y=yt_arma2,x=yt_2_arma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-2}\).

Yt dengan Yt-3
#Yt-3
yt_3_arma2 <- arma22[1:297]
yt_3_arma2
##   [1] -1.3696174977 -1.7982846779 -3.7564803134 -4.1844989443 -2.4380580230
##   [6] -2.0573868423 -0.0910098080 -1.4490056726 -0.6076282073 -1.3496925124
##  [11] -1.0829677579 -1.7023873257 -2.5197440501 -2.7197994281 -2.1967896169
##  [16] -1.7916785770 -0.4348298378 -1.0473867239 -0.3192377354 -0.3969101190
##  [21] -0.6227524142 -0.0924324177  0.9325752668  3.0122584478  2.4454839633
##  [26]  1.4760437919 -0.0643674725 -0.5461726632 -0.6784655769  0.4243199563
##  [31]  1.9301392423  2.1248949046  3.9725615519  4.5550448477  5.3150026476
##  [36]  5.6203880642  4.2409849415  4.1294892573  0.9252426580  1.3701387371
##  [41] -0.5288919768 -0.5929594395 -0.9757027529 -0.5211792743 -0.5538914045
##  [46] -0.9307075065 -0.0187023435 -0.6367046061 -2.2069303322 -2.6390626356
##  [51] -3.4043427015 -2.4297860877  0.1390567838  1.6579727568  0.8637556594
##  [56]  2.5428452215  0.6022881183  2.1723143763 -0.2438956571 -0.5951906827
##  [61] -2.9849387954 -1.7339822802 -4.6654646818 -3.7836495546 -3.9666681459
##  [66] -4.1325902763 -3.0262664639 -4.2864109226 -3.7622722444 -4.7151923322
##  [71] -4.5967456399 -3.0173218097 -4.2835223541 -3.3062941169 -3.5092919106
##  [76] -2.6374499698 -1.5337718084 -1.3312387130  0.7849170050  1.6064013034
##  [81]  3.9292469905  1.9596439795  0.9664737813  0.0669866797 -0.1584849418
##  [86] -0.4141202333 -0.6312709023 -0.4620127937  0.4949622808 -0.3416039498
##  [91] -0.9383995764 -1.4427075147 -2.2610634558 -2.1325223511 -1.1874697279
##  [96] -1.5071768803  0.3218529332 -1.3783213458  0.3085934849  2.4311778976
## [101]  2.0806680196  2.9673723690  1.4530768428  1.9792495800  2.3624139896
## [106]  1.8422568201  1.7614003802  1.9584343073  2.7174696443  2.5529423211
## [111]  1.4754614153  1.3554392078  0.9762882119  2.1519800469  2.0506378942
## [116]  1.0566362664 -0.1564765121 -0.6963019933 -2.2441924588 -1.8416334056
## [121] -1.8481320229 -0.9362785485  1.0132172775 -0.1475489089 -1.0352244145
## [126] -0.4098905792  0.7560315697  3.5576045354  2.1448787201  1.8165941197
## [131] -0.5235159857 -1.1697917262 -1.5151996272 -0.2480426055  0.1076470164
## [136] -0.0009816395  0.2239496072 -0.3267270455  0.9924955244 -0.8552703690
## [141]  0.6436579300  1.2226656780  3.0756483638  2.7230731085  1.1818983317
## [146] -0.2938174133  0.2347314183 -1.1966072807  0.3867847283 -1.3827057885
## [151] -0.7294145525 -0.9784002367 -1.2956348622  0.0031996111 -0.3001120910
## [156]  1.7887425967  3.7384623212  2.3458004938  4.0714796600  2.7815006616
## [161]  3.1478649220  1.6664269995  3.2494060618  2.6280455760  5.4679647463
## [166]  3.5571011493  4.5485533388  4.9684033961  5.0280773241  4.6975450886
## [171]  2.6840203628  1.6290009627  0.1999273113  0.5740551176 -0.4256382849
## [176]  0.5406114019  0.7418567933  1.2997851726  1.6025208730  1.6136129632
## [181]  2.2928430253  1.4468904019  1.8612868351  0.8777139291 -0.0610314965
## [186] -1.3332268089 -0.9125263389  0.3581329351 -0.7915369668  0.4955112665
## [191] -0.4502244253  1.4675554008  3.0521765941  2.0408339742  3.8704090143
## [196]  3.9605441093  3.9360503187  3.5445138311  0.5049588491  0.3915050022
## [201] -2.8725640338 -2.0091667120 -3.0173010664 -1.8381977156 -1.5748868384
## [206] -1.2514830640 -2.5783738178 -3.7484191100 -3.8057155958 -3.0969262848
## [211] -1.4586533030  0.1531135428  0.8236045641 -0.3790786936 -1.0001442542
## [216] -2.2513926710 -2.5717361239  0.1246023618 -0.5437266688  2.9508487574
## [221]  1.5184775014  2.4469359872  3.3434532407  3.1174139183  2.9978533478
## [226]  2.4173578196  1.5319233546  1.7966371548  1.2212161425  1.9888356266
## [231]  2.5393974322  3.7546275738  1.3043079574 -0.2394983994 -1.7460452432
## [236] -0.7708653105 -0.3849718038 -1.1442085530 -1.5138630876 -2.1968083475
## [241] -1.6095317523 -0.9686538607 -1.8729730474 -2.5265869712 -3.1503634658
## [246] -1.6735889734 -1.2527046715 -0.3854214991 -1.2299702469 -1.3664692859
## [251] -1.3620096392 -0.8698815763 -1.1048884660 -2.4558106193 -3.2188795522
## [256] -5.4245906442 -5.7588922841 -5.9623854154 -5.0454539055 -3.9564322618
## [261] -3.7899616180 -5.3313166036 -4.8885674466 -2.0508739512 -2.0599004072
## [266] -1.0821178750  0.1666168226 -0.8877977615  0.8502273032 -0.1476428826
## [271] -0.4367626603 -2.1069453839 -4.5465410226 -4.8087456252 -4.2289076874
## [276] -1.9178765546 -2.2804864851 -0.4098672855 -1.6178574433 -0.0137269804
## [281]  0.0122011943  1.4282807605  0.6560697941  0.7567608782 -0.8228605730
## [286] -1.1860435575 -1.8041134735 -0.4466562647 -0.3954967938  2.1487643901
## [291]  0.9586140341  2.2653125680  0.2588626762  1.0329576284 -0.9297517508
## [296] -1.2122608974 -2.3845725749
plot(y=yt_arma2,x=yt_3_arma2)

Berdasarkan scatterplot tersebut, terlihat bahwa terdapat hubungan positif antara \(Y_t\) dan \(Y_{t-3}\).

Korelasi Antar Lag

Data Bangkitan
cor(yt_arma2, yt_1_arma2, use = "complete.obs")
## [1] 0.8628549

Nilai korelasi antara \(Y_t\) dengan \(Y_{t-1}\) dari hasil simulasi cukup mendekati dengan teori, yaitu:

\[ \rho_1 = \frac{(1 - \phi_2)\theta_1 + (\phi_1 - \phi_2)\theta_2 + \phi_1(1 + \theta_1^2 + \theta_2^2)} {(1 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + 2(\phi_1 - \phi_2)\theta_1\theta_2} = \frac{(1 - 0.2)(0.4) + (0.5 - 0.2)(0.6) + 0.5(1 + 0.4^2 + 0.6^2)} {(1 - 0.2)(1 + 0.4^2 + 0.6^2) + 2(0.5 - 0.2)(0.4)(0.6)} = \frac{1.26}{1.36} \approx 0.926 \]

cor(yt_arma2, yt_2_arma2, use = "complete.obs")
## [1] 0.7616306

Nilai korelasi antara \(Y_t\) dengan \(Y_{t-2}\) dari hasil simulasi cukup mendekati dengan teori, yaitu:

\[ \rho_2 = \frac{(\phi_1 + \theta_1\theta_2)(1 - \phi_2) + (\phi_1^2 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + \theta_2(\theta_1 + \phi_1\theta_2)} {(1 - \phi_2)(1 + \theta_1^2 + \theta_2^2) + 2(\phi_1 - \phi_2)\theta_1\theta_2} = \frac{(0.5 + 0.4 \times 0.6)(0.8) + (0.5^2 - 0.2)(1 + 0.4^2 + 0.6^2) + 0.6(0.4 + 0.5 \times 0.6)} {(1-0.2)(1 + 0.4^2 + 0.6^2) + 2(0.5 - 0.2)(0.4)(0.6)} = \frac{1.088}{1.36} = 0.8 \]

cor(yt_arma2, yt_3_arma2, use = "complete.obs")
## [1] 0.5721462

Nilai korelasi antara \(Y_t\) dengan \(Y_{t-3}\) dari hasil simulasi cukup mendekati teori, yaitu:

\[ \rho_3 = \phi_1 \rho_2 + \phi_2 \rho_1 = (0.5)(0.8) + (0.2)(0.926) = 0.585 \]

Kesimpulan: Meskipun EACF mengindikasikan ARMA(1,2), pola ACF dan PACF yang tails off serta nilai autokorelasi empiris dengan nilai teoritis yang tidak berbeda jauh menunjukkan bahwa model ARMA(2,2) merupakan representasi yang konsisten untuk proses bangkitan. Perbedaan kecil pada hasil identifikasi tampaknya muncul akibat variabilitas sampel dan sensitivitas EACF, bukan karena inkonsistensi struktural model.

Zifa Aura Rahman
G1401231010